HVT Scoring Cells with Layers using scoreLayeredHVT

Zubin Dowlaty, Srinivasan Sudarsanam, Somya Shambhawi

2024-01-31

1 Abstract

The HVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data analysis. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

  1. Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.

  2. Data Projection: Dimension projection of the compressed cells to 1D,2D or Interactive Surface plot with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension.

  3. Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.

  4. Scoring: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

2 Example : HVT with the Torus dataset

In this section, we will see how we can use the package to visualize multidimensional data by projecting them to two dimensions using Sammon’s projection and further used for Scoring

Data Understanding

First of all, let us see how to generate data for torus. We are using a library geozoo for this purpose. Geo Zoo (stands for Geometric Zoo) is a compilation of geometric objects ranging from three to 10 dimensions. Geo Zoo contains regular or well-known objects, eg cube and sphere, and some abstract objects, e.g. Boy’s surface, Torus and Hyper-Torus.

Here, we will generate a 3D torus (a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle) with 9000 points.

Raw Torus Dataset

The torus dataset includes the following columns:

Lets, explore the raw torus dataset containing 12000 points. For the sake of brevity we are displaying first 6 rows.

set.seed(240)
# Here p represents dimension of object
# n represents number of points
torus <- geozoo::torus(p = 3,n = 12000)
torus_df <- data.frame(torus$points)
colnames(torus_df) <- c("x","y","z")
torus_df <- torus_df %>% round(4)
Table(head(torus_df), scroll = TRUE, limit = 20)
x y z
-2.6282 0.5656 -0.7253
-1.4179 -0.8903 0.9455
-1.0308 1.1066 -0.8731
1.8847 0.1895 0.9944
-1.9506 -2.2507 0.2071
-1.4824 0.9229 0.9672

Now, let us check the structure of the data and analyse its summary.

str(torus_df)
#> 'data.frame':    12000 obs. of  3 variables:
#>  $ x: num  -2.63 -1.42 -1.03 1.88 -1.95 ...
#>  $ y: num  0.566 -0.89 1.107 0.19 -2.251 ...
#>  $ z: num  -0.725 0.946 -0.873 0.994 0.207 ...
summary(torus_df)
#>        x                   y                  z            
#>  Min.   :-2.997700   Min.   :-2.99930   Min.   :-1.000000  
#>  1st Qu.:-1.149025   1st Qu.:-1.11332   1st Qu.:-0.711950  
#>  Median :-0.007000   Median : 0.01305   Median : 0.015300  
#>  Mean   :-0.001444   Mean   : 0.01035   Mean   : 0.004423  
#>  3rd Qu.: 1.140325   3rd Qu.: 1.13373   3rd Qu.: 0.718550  
#>  Max.   : 2.999500   Max.   : 2.99930   Max.   : 1.000000

Let us first split the data into train and test. We will randomly select 80% of the data for training and remaining as testing.


num_rows <- nrow(torus_df)
set.seed(123)
train_indices <- sample(1:num_rows, 0.8 * num_rows)
traindata <- torus_df[train_indices, ]
testdata <- torus_df[-train_indices, ]

Training Dataset

Now, lets have a look at the randomly selected training dataset containing (9600 data points). For the sake of brevity we are displaying first six rows.

row.names(traindata) <- NULL
Table(head(traindata))
x y z
1.4935 -1.6012 -0.9819
-2.8712 0.3054 0.4609
-1.6231 1.4452 -0.9849
-0.7097 -2.7012 -0.6094
-0.4525 1.1998 0.6964
0.1615 -1.2518 -0.6750

Testing Dataset

Now, lets have a look at the randomly selected testing dataset containing (2400 data points). For the sake of brevity we are displaying first six rows.

rownames(testdata) <- NULL
Table(head(testdata))
x y z
-1.0308 1.1066 -0.8731
1.8847 0.1895 0.9944
-1.0046 -1.8170 -0.9971
-2.4446 -1.6528 0.3097
-0.3961 -2.1775 0.9770
-1.1130 -0.6516 -0.7040

3 Map A : Base Compressed Map

Let us try to visualize the compressed Map A from the flow diagram below.

Figure 1: Data Segregation with highlighted bounding box in red around compressed map A

Figure 1: Data Segregation with highlighted bounding box in red around compressed map A

This package can perform vector quantization using the following algorithms -

For more information on vector quantization, refer the following link.

The trainHVT function constructs highly compressed hierarchical Voronoi tessellations. The raw data is first scaled and this scaled data is supplied as input to the vector quantization algorithm. The vector quantization algorithm compresses the dataset until a user-defined compression percentage/rate is achieved using a parameter called quantization error which acts as a threshold and determines the compression percentage. It means that for a given user-defined compression percentage we get the ‘n’ number of cells, then all of these cells formed will have a quantization error less than the threshold quantization error.

Let’s try to comprehend the trainHVT function first before moving ahead.

trainHVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  normalize = TRUE,
  diagnose = FALSE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters of trainHVT function have been explained below:

The output of trainHVT function (list of 6 elements) have been explained below:

We will use the trainHVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the trainHVT function again.

As this is already done in HVT Vignette: please refer for more information.

Model Parameters

set.seed(240)
torus_mapA <- trainHVT(
  traindata,
  n_cells = 900,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = FALSE,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s check the compression summary for torus.

compressionSummaryTable(torus_mapA[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 900 751 0.83 n_cells: 900 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

We successfully compressed 83% of the data using n_cells parameter as 900, the next step involves performing data projection on the compressed data. In this step, the compressed data will be transformed and projected onto a lower-dimensional space to visualize and analyze the data in a more manageable form.

As per the manual, torus_mapA[[3]] gives us detailed information about the hierarchical vector quantized data. torus_mapA[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from torus_mapA showing Cell.ID, Centroids and Quantization Error for each of the 900 cells.

summaryTable(torus_mapA[[3]]$summary,scroll = TRUE,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error x y z
1 1 1 9 711 0.11 -2.84 0.62 -0.39
1 1 2 17 579 0.1 1.16 0.89 0.84
1 1 3 13 493 0.09 -2.04 0.03 -1.00
1 1 4 9 503 0.07 2.85 0.43 -0.47
1 1 5 7 241 0.05 0.02 -1.27 0.68
1 1 6 13 787 0.11 -2.59 1.15 0.53
1 1 7 14 736 0.11 1.83 1.54 0.91
1 1 8 8 159 0.08 -2.41 -1.35 0.65
1 1 9 13 11 0.12 1.65 -2.45 0.29
1 1 10 7 261 0.05 -0.58 -1.15 -0.70
1 1 11 11 684 0.08 -0.63 1.29 0.83
1 1 12 11 170 0.06 -0.95 -1.68 1.00
1 1 13 14 109 0.12 -2.55 -1.56 0.06
1 1 14 12 443 0.09 -2.22 -0.27 0.97
1 1 15 10 631 0.08 -2.46 0.44 0.86
1 1 16 16 296 0.07 -1.69 -0.85 -0.99
1 1 17 9 826 0.11 2.16 1.98 0.35
1 1 18 16 385 0.08 -1.80 -0.54 -0.99
1 1 19 10 432 0.06 -0.97 -0.25 0.08
1 1 20 10 810 0.08 -1.30 2.00 0.92
1 1 21 6 90 0.06 -0.16 -2.28 -0.96
1 1 22 6 21 0.09 -0.57 -2.92 0.20
1 1 23 9 383 0.1 -2.57 -0.58 0.77
1 1 24 10 160 0.1 -2.62 -1.25 0.41
1 1 25 9 338 0.06 0.95 -0.71 -0.57
1 1 26 11 292 0.06 0.15 -1.01 0.21
1 1 27 12 528 0.07 -1.93 0.23 -0.99
1 1 28 13 781 0.08 0.11 2.03 -1.00
1 1 29 15 197 0.12 -2.83 -0.95 0.10
1 1 30 9 576 0.04 0.49 0.93 -0.30
1 1 31 9 852 0.09 -0.06 2.62 0.78
1 1 32 11 246 0.07 -0.19 -1.24 -0.66
1 1 33 13 440 0.05 -1.12 -0.21 0.51
1 1 34 8 290 0.09 2.31 -0.47 -0.93
1 1 35 13 193 0.09 0.76 -1.40 -0.91
1 1 36 13 599 0.08 -1.94 0.51 -1.00
1 1 37 11 870 0.09 0.91 2.80 -0.30
1 1 38 8 272 0.05 1.16 -0.89 0.84
1 1 39 7 32 0.07 -0.24 -2.81 -0.56
1 1 40 9 280 0.05 0.00 -1.08 0.38
1 1 41 11 333 0.12 2.91 -0.29 0.35
1 1 42 14 877 0.08 1.19 2.74 -0.12
1 1 43 14 526 0.07 0.90 0.64 0.46
1 1 44 16 507 0.08 0.93 0.51 0.36
1 1 45 8 504 0.06 1.26 0.50 -0.76
1 1 46 14 498 0.06 -1.05 0.25 -0.38
1 1 47 16 464 0.09 -1.55 -0.12 0.89
1 1 48 9 682 0.08 -0.06 1.35 0.76
1 1 49 14 373 0.08 -1.22 -0.61 -0.77
1 1 50 10 554 0.09 2.72 0.60 -0.61
1 1 51 9 720 0.08 -0.54 1.52 0.92
1 1 52 10 81 0.09 1.20 -2.16 -0.88
1 1 53 15 780 0.1 1.72 1.83 0.85
1 1 54 9 640 0.06 0.16 1.18 0.58
1 1 55 15 480 0.08 -1.90 -0.02 0.99
1 1 56 6 339 0.06 1.13 -0.66 0.72
1 1 57 6 790 0.05 0.77 2.16 0.95
1 1 58 10 205 0.09 2.91 -0.56 0.27
1 1 59 10 104 0.07 -1.35 -2.11 -0.86
1 1 60 7 153 0.07 -0.58 -1.77 -0.99
1 1 61 6 768 0.05 -0.18 1.92 1.00
1 1 62 9 250 0.06 -0.81 -1.19 -0.83
1 1 63 7 572 0.06 0.46 0.90 -0.13
1 1 64 12 365 0.07 0.86 -0.56 0.25
1 1 65 10 580 0.07 -0.71 0.82 -0.40
1 1 66 13 801 0.13 2.18 1.85 -0.49
1 1 67 8 454 0.09 -2.96 -0.23 0.23
1 1 68 9 243 0.05 -1.47 -1.09 -0.99
1 1 69 13 860 0.12 1.81 2.38 0.03
1 1 70 9 844 0.07 -0.33 2.54 0.83
1 1 71 6 157 0.07 -0.84 -1.74 -1.00
1 1 72 14 435 0.06 1.27 0.00 0.68
1 1 73 18 714 0.12 1.42 1.49 0.99
1 1 74 12 402 0.06 1.17 -0.27 -0.60
1 1 75 8 749 0.09 -2.30 1.07 0.84
1 1 76 9 716 0.1 1.72 1.50 -0.95
1 1 77 10 421 0.1 2.63 0.04 0.77
1 1 78 7 180 0.05 -1.95 -1.40 0.92
1 1 79 12 721 0.09 2.73 1.20 -0.16
1 1 80 16 712 0.08 -2.08 0.94 0.96
1 1 81 6 91 0.08 -0.85 -2.30 -0.89
1 1 82 11 332 0.07 0.55 -0.83 -0.04
1 1 83 13 441 0.06 1.04 0.04 -0.30
1 1 84 6 539 0.06 1.45 0.68 -0.91
1 1 85 11 789 0.08 -2.38 1.30 -0.70
1 1 86 7 626 0.07 -0.11 1.12 -0.49
1 1 87 11 354 0.07 -0.74 -0.77 -0.35
1 1 88 10 368 0.1 -2.89 -0.56 0.31
1 1 89 11 123 0.08 -1.60 -1.85 -0.89
1 1 90 7 77 0.05 -1.17 -2.33 -0.79
1 1 91 12 142 0.07 -1.02 -1.91 0.98
1 1 92 5 195 0.06 -0.33 -1.56 0.91
1 1 93 13 524 0.1 1.06 0.62 0.63
1 1 94 18 637 0.07 1.57 1.04 0.99
1 1 95 8 135 0.06 -1.81 -1.73 0.86
1 1 96 17 895 0.14 -1.36 2.62 0.30
1 1 97 15 848 0.16 -2.46 1.62 0.31
1 1 98 12 627 0.06 0.51 1.12 0.64
1 1 99 6 582 0.09 -2.86 0.20 0.48
1 1 100 12 552 0.11 2.58 0.57 0.76
1 1 101 12 515 0.07 -0.93 0.39 0.16
1 1 102 16 899 0.13 -1.55 2.51 -0.26
1 1 103 8 30 0.09 0.59 -2.72 -0.62
1 1 104 13 143 0.11 -2.70 -1.29 0.05
1 1 105 10 417 0.06 1.08 -0.15 -0.42
1 1 106 11 210 0.06 1.24 -1.18 0.96
1 1 107 5 306 0.04 -0.15 -0.99 0.08
1 1 108 11 565 0.07 -1.39 0.57 -0.86
1 1 109 11 624 0.05 0.24 1.14 -0.55
1 1 110 8 155 0.05 1.91 -1.25 -0.96
1 1 111 12 618 0.07 0.73 1.12 -0.75
1 1 112 11 102 0.09 -1.96 -1.96 0.63
1 1 113 10 670 0.1 -2.69 0.48 0.67
1 1 114 6 238 0.04 1.63 -0.87 0.99
1 1 115 20 422 0.15 -2.09 -0.31 -0.99
1 1 116 8 568 0.08 1.97 0.77 -0.99
1 1 117 10 657 0.04 0.06 1.24 0.65
1 1 118 5 288 0.03 -0.19 -1.06 0.38
1 1 119 12 323 0.07 1.98 -0.50 -1.00
1 1 120 5 632 0.06 -0.22 1.12 0.51
1 1 121 15 172 0.09 2.56 -0.89 -0.70
1 1 122 7 897 0.1 -0.23 2.98 0.07
1 1 123 9 510 0.06 -1.40 0.21 0.81
1 1 124 6 95 0.07 1.72 -1.85 -0.85
1 1 125 9 424 0.07 1.70 -0.02 -0.95
1 1 126 15 372 0.05 1.20 -0.42 0.68
1 1 127 10 818 0.11 -2.67 1.30 0.25
1 1 128 13 94 0.11 -2.32 -1.81 0.30
1 1 129 12 797 0.09 0.72 2.22 -0.94
1 1 130 9 745 0.08 2.18 1.53 -0.74
1 1 131 11 60 0.08 -0.25 -2.59 -0.80
1 1 132 13 532 0.09 -1.09 0.49 -0.59
1 1 133 11 264 0.08 0.61 -1.05 0.62
1 1 134 13 643 0.09 0.37 1.25 -0.72
1 1 135 17 884 0.13 -2.18 2.02 0.17
1 1 136 12 531 0.07 -1.33 0.38 0.79
1 1 137 6 287 0.05 -0.71 -1.03 -0.67
1 1 138 11 541 0.07 0.92 0.73 -0.56
1 1 139 15 837 0.12 1.82 2.20 0.51
1 1 140 13 55 0.1 2.27 -1.86 0.34
1 1 141 18 630 0.07 1.33 1.07 0.95
1 1 142 10 704 0.06 -0.35 1.44 0.85
1 1 143 8 304 0.06 -0.35 -1.00 -0.34
1 1 144 12 587 0.09 0.62 0.95 0.50
1 1 145 12 298 0.05 -0.59 -1.01 -0.55
1 1 146 9 659 0.12 -2.82 0.38 -0.52
1 1 147 11 828 0.11 -2.66 1.38 0.03
1 1 148 9 218 0.06 -0.80 -1.35 -0.90
1 1 149 9 233 0.08 0.98 -1.13 -0.86
1 1 150 16 567 0.08 -0.68 0.75 0.17
1 1 151 9 364 0.08 2.61 -0.23 0.78
1 1 152 9 49 0.06 -1.17 -2.58 -0.55
1 1 153 7 458 0.08 2.40 0.18 0.91
1 1 154 14 851 0.09 -0.02 2.62 -0.78
1 1 155 9 377 0.06 0.88 -0.50 -0.15
1 1 156 8 680 0.05 -0.49 1.33 -0.81
1 1 157 12 616 0.08 -0.58 0.98 0.51
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1 1 159 8 17 0.07 0.21 -2.90 0.42
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1 1 161 12 192 0.07 1.15 -1.29 -0.96
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1 1 177 10 506 0.06 0.87 0.52 0.13
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1 1 246 12 841 0.12 2.13 2.10 0.09
1 1 247 8 562 0.04 0.66 0.85 -0.38
1 1 248 12 543 0.09 1.37 0.68 0.88
1 1 249 10 412 0.09 2.90 0.00 0.42
1 1 250 11 64 0.11 -0.71 -2.63 0.68
1 1 251 8 646 0.05 0.81 1.26 -0.86
1 1 252 10 585 0.05 0.34 0.97 -0.23
1 1 253 12 694 0.08 1.18 1.42 0.99
1 1 254 7 674 0.09 2.73 0.95 -0.44
1 1 255 8 889 0.08 0.14 2.96 0.25
1 1 256 12 760 0.08 2.15 1.56 0.75
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1 1 259 21 559 0.12 1.75 0.70 0.99
1 1 260 11 625 0.08 1.85 0.98 -0.99
1 1 261 9 629 0.06 -1.54 0.78 -0.96
1 1 262 12 386 0.07 1.85 -0.25 -0.99
1 1 263 7 647 0.07 -1.77 0.69 0.99
1 1 264 15 609 0.06 0.07 1.03 0.26
1 1 265 10 537 0.08 -2.25 0.20 -0.96
1 1 266 13 766 0.06 -1.58 1.54 0.98
1 1 267 14 146 0.08 1.70 -1.43 -0.97
1 1 268 10 621 0.05 0.06 1.08 0.40
1 1 269 15 52 0.11 0.97 -2.54 0.69
1 1 270 7 12 0.06 0.65 -2.86 -0.35
1 1 271 12 509 0.06 -1.00 0.33 0.31
1 1 272 13 880 0.1 -1.05 2.64 -0.52
1 1 273 4 563 0.03 0.58 0.85 -0.25
1 1 274 7 785 0.06 -0.49 2.01 -1.00
1 1 275 12 802 0.1 2.05 1.87 0.62
1 1 276 11 270 0.06 0.88 -0.99 0.74
1 1 277 12 550 0.06 -0.79 0.62 -0.08
1 1 278 7 10 0.08 0.01 -2.99 0.07
1 1 279 10 101 0.09 2.66 -1.38 0.02
1 1 280 10 871 0.08 1.16 2.71 0.30
1 1 281 10 754 0.11 0.70 1.88 -0.99
1 1 282 9 825 0.08 -1.81 1.86 0.80
1 1 283 15 561 0.07 0.84 0.81 0.55
1 1 284 10 237 0.07 -0.47 -1.32 0.80
1 1 285 16 660 0.08 1.05 1.22 0.92
1 1 286 10 540 0.05 0.71 0.72 -0.13
1 1 287 10 776 0.11 2.52 1.55 0.24
1 1 288 11 717 0.05 -0.60 1.56 -0.94
1 1 289 8 167 0.08 -1.20 -1.67 0.99
1 1 290 8 471 0.06 1.04 0.29 0.38
1 1 291 7 36 0.07 -1.37 -2.60 -0.33
1 1 292 14 642 0.07 -0.30 1.18 -0.62
1 1 293 11 2 0.12 1.84 -2.36 0.02
1 1 294 12 449 0.07 -1.52 -0.14 -0.88
1 1 295 11 106 0.09 1.98 -1.63 -0.82
1 1 296 11 875 0.11 -0.77 2.69 0.59
1 1 297 12 48 0.11 -1.80 -2.32 -0.33
1 1 298 16 42 0.1 -1.34 -2.61 0.34
1 1 299 20 16 0.13 1.23 -2.63 0.41
1 1 300 8 575 0.07 -1.75 0.42 0.98
1 1 301 10 713 0.1 2.58 1.24 -0.51
1 1 302 12 900 0.12 -1.19 2.74 0.00
1 1 303 9 248 0.05 0.01 -1.20 -0.60
1 1 304 10 284 0.05 0.01 -1.05 -0.30
1 1 305 8 345 0.08 1.83 -0.42 0.99
1 1 306 8 45 0.09 2.11 -2.00 -0.41
1 1 307 9 232 0.07 0.21 -1.30 0.73
1 1 308 9 740 0.06 -1.65 1.30 0.99
1 1 309 6 279 0.05 -0.41 -1.07 -0.52
1 1 310 12 163 0.09 -0.32 -1.73 -0.97
1 1 311 17 108 0.11 -0.52 -2.15 -0.97
1 1 312 13 434 0.05 -1.03 -0.26 0.35
1 1 313 26 257 0.1 1.55 -0.81 -0.96
1 1 314 14 78 0.1 -0.46 -2.48 0.85
1 1 315 8 607 0.04 0.34 1.07 -0.47
1 1 316 13 833 0.1 2.04 2.12 -0.32
1 1 317 8 514 0.06 -2.18 0.07 0.98
1 1 318 19 876 0.13 -2.19 1.93 -0.36
1 1 319 14 401 0.1 2.26 -0.13 -0.96
1 1 320 16 874 0.12 -0.30 2.77 -0.61
1 1 321 11 398 0.07 1.30 -0.26 -0.74
1 1 322 9 390 0.06 1.05 -0.36 -0.46
1 1 323 10 367 0.06 1.00 -0.50 0.47
1 1 324 9 809 0.08 -1.58 1.85 0.90
1 1 325 10 396 0.09 -1.44 -0.52 0.88
1 1 326 8 253 0.06 0.56 -1.13 -0.67
1 1 327 5 792 0.03 -0.45 2.07 0.99
1 1 328 14 156 0.07 0.39 -1.73 0.97
1 1 329 13 231 0.08 2.04 -0.75 -0.98
1 1 330 5 499 0.03 1.96 0.40 -1.00
1 1 331 14 842 0.08 -0.88 2.42 0.82
1 1 332 7 260 0.06 -0.17 -1.19 0.60
1 1 333 9 683 0.12 -2.89 0.46 0.36
1 1 334 15 314 0.06 0.38 -0.93 0.05
1 1 335 8 574 0.05 0.79 0.92 -0.61
1 1 336 11 533 0.05 -0.85 0.53 0.06
1 1 337 16 752 0.09 1.04 1.86 -0.99
1 1 338 9 691 0.07 -2.00 0.89 -0.98
1 1 339 7 813 0.06 0.16 2.28 0.96
1 1 340 12 459 0.05 1.01 0.15 0.19
1 1 341 12 151 0.07 1.17 -1.55 -1.00
1 1 342 19 436 0.07 -1.36 -0.28 0.79
1 1 343 16 551 0.07 -0.87 0.61 -0.35
1 1 344 9 378 0.06 0.89 -0.48 0.16
1 1 345 10 265 0.05 0.13 -1.10 -0.46
1 1 346 18 494 0.09 1.23 0.40 0.71
1 1 347 9 710 0.05 0.55 1.55 0.94
1 1 348 3 8 0.07 0.36 -2.93 -0.28
1 1 349 7 9 0.07 0.14 -2.96 -0.27
1 1 350 9 315 0.03 -0.37 -0.98 0.31
1 1 351 12 226 0.07 0.67 -1.24 -0.81
1 1 352 10 648 0.08 -1.76 0.75 -0.99
1 1 353 11 293 0.06 -0.10 -1.04 0.28
1 1 354 22 693 0.08 0.75 1.53 -0.95
1 1 355 8 150 0.08 -0.07 -1.76 -0.97
1 1 356 9 212 0.04 1.65 -1.00 1.00
1 1 357 11 556 0.07 -2.62 0.16 0.78
1 1 358 8 5 0.07 0.37 -2.97 0.10
1 1 359 11 463 0.06 -1.17 -0.06 0.55
1 1 360 13 351 0.06 -0.61 -0.79 -0.02
1 1 361 8 173 0.06 1.22 -1.44 0.99
1 1 362 13 33 0.09 -0.86 -2.80 -0.35
1 1 363 8 356 0.06 0.79 -0.66 -0.24
1 1 364 14 69 0.11 -1.64 -2.31 0.55
1 1 365 7 779 0.08 2.32 1.63 0.54
1 1 366 15 128 0.11 -2.18 -1.67 0.66
1 1 367 11 25 0.09 0.01 -2.84 -0.53
1 1 368 8 220 0.05 -0.58 -1.36 -0.85
1 1 369 7 249 0.08 1.22 -0.97 -0.90
1 1 370 13 816 0.12 -1.25 2.08 -0.90
1 1 371 4 254 0.07 -0.43 -1.20 -0.69
1 1 372 6 273 0.06 1.42 -0.77 0.92
1 1 373 9 661 0.06 -0.66 1.18 -0.76
1 1 374 10 611 0.04 -1.30 0.76 -0.87
1 1 375 7 795 0.06 -0.36 2.13 -0.99
1 1 376 12 692 0.07 2.82 1.02 0.08
1 1 377 14 86 0.11 1.41 -2.01 -0.88
1 1 378 12 251 0.07 -1.38 -1.14 0.98
1 1 379 8 517 0.05 -1.01 0.39 -0.39
1 1 380 13 119 0.1 0.92 -1.89 -0.99
1 1 381 13 240 0.06 0.25 -1.23 -0.67
1 1 382 12 140 0.07 0.62 -1.84 1.00
1 1 383 10 363 0.07 1.09 -0.53 -0.62
1 1 384 13 357 0.08 0.75 -0.67 -0.03
1 1 385 13 46 0.11 1.52 -2.33 0.62
1 1 386 5 723 0.11 -2.53 0.82 -0.74
1 1 387 10 725 0.07 2.61 1.21 0.47
1 1 388 12 885 0.11 0.31 2.94 -0.28
1 1 389 11 465 0.09 2.63 0.21 -0.76
1 1 390 16 439 0.08 -2.49 -0.30 0.86
1 1 391 10 374 0.06 -0.77 -0.64 0.05
1 1 392 8 62 0.08 -1.17 -2.57 0.55
1 1 393 13 853 0.09 1.02 2.62 0.58
1 1 394 15 278 0.07 -0.13 -1.07 -0.38
1 1 395 9 746 0.06 -0.30 1.71 0.96
1 1 396 13 544 0.07 -0.86 0.58 0.26
1 1 397 10 176 0.11 2.72 -0.80 0.53
1 1 398 8 235 0.07 -0.17 -1.32 0.74
1 1 399 10 671 0.1 -2.60 0.52 -0.76
1 1 400 13 588 0.08 -1.03 0.75 0.68
1 1 401 10 117 0.08 0.26 -2.09 0.99
1 1 402 15 187 0.09 1.74 -1.09 -1.00
1 1 403 12 846 0.11 1.54 2.41 -0.50
1 1 404 8 817 0.08 0.58 2.34 0.91
1 1 405 6 788 0.07 -1.38 1.82 -0.96
1 1 406 5 605 0.03 0.21 1.06 -0.39
1 1 407 11 516 0.07 -0.93 0.40 -0.19
1 1 408 9 523 0.05 0.83 0.62 -0.25
1 1 409 6 202 0.05 -1.01 -1.43 -0.97
1 1 410 5 416 0.04 0.98 -0.19 0.00
1 1 411 12 408 0.07 -1.23 -0.46 0.73
1 1 412 7 475 0.07 -1.35 0.04 0.76
1 1 413 5 814 0.06 0.35 2.30 0.94
1 1 414 7 207 0.06 -0.63 -1.46 0.91
1 1 415 10 733 0.08 -2.19 1.08 -0.89
1 1 416 11 867 0.07 0.28 2.77 0.62
1 1 417 12 468 0.12 1.58 0.23 0.91
1 1 418 11 474 0.08 1.37 0.31 -0.80
1 1 419 8 113 0.12 2.53 -1.29 -0.54
1 1 420 7 615 0.05 0.11 1.10 -0.44
1 1 421 10 722 0.06 0.04 1.60 0.92
1 1 422 13 482 0.07 -1.00 0.15 -0.16
1 1 423 9 97 0.09 1.03 -2.09 -0.94
1 1 424 16 222 0.07 0.36 -1.30 -0.76
1 1 425 9 201 0.07 2.41 -0.78 0.84
1 1 426 10 858 0.13 -1.67 2.17 0.66
1 1 427 10 426 0.05 -0.97 -0.28 -0.15
1 1 428 8 329 0.05 -0.39 -0.92 0.00
1 1 429 10 125 0.08 0.00 -1.98 -1.00
1 1 430 13 376 0.08 -1.91 -0.62 1.00
1 1 431 17 388 0.08 -1.70 -0.56 0.98
1 1 432 6 92 0.06 0.05 -2.34 0.94
1 1 433 10 141 0.07 -0.36 -1.93 1.00
1 1 434 11 490 0.1 1.44 0.37 0.86
1 1 435 7 774 0.05 -0.43 1.92 1.00
1 1 436 13 491 0.09 -1.64 0.08 -0.93
1 1 437 4 782 0.04 -1.86 1.48 0.92
1 1 438 8 330 0.04 -0.51 -0.92 0.32
1 1 439 14 850 0.08 0.46 2.63 0.74
1 1 440 15 610 0.06 0.95 1.06 -0.81
1 1 441 7 639 0.08 2.57 0.86 0.70
1 1 442 2 289 0.02 -0.39 -1.06 0.50
1 1 443 12 739 0.06 -1.94 1.18 0.96
1 1 444 17 602 0.09 1.17 1.02 -0.89
1 1 445 12 400 0.09 -2.95 -0.45 -0.16
1 1 446 8 127 0.1 -0.84 -2.08 0.97
1 1 447 5 676 0.05 -0.84 1.23 -0.86
1 1 448 7 41 0.1 -1.73 -2.42 0.18
1 1 449 15 425 0.09 -1.73 -0.29 -0.97
1 1 450 15 888 0.11 -1.58 2.43 0.44
1 1 451 9 217 0.07 2.36 -0.71 -0.88
1 1 452 8 771 0.08 1.30 1.91 0.95
1 1 453 9 859 0.1 -1.22 2.43 0.69
1 1 454 15 311 0.1 -1.98 -0.80 -0.99
1 1 455 15 230 0.08 -2.43 -0.89 -0.80
1 1 456 11 703 0.08 0.28 1.50 0.88
1 1 457 8 672 0.06 2.82 0.93 -0.23
1 1 458 5 182 0.06 -1.74 -1.41 0.97
1 1 459 12 100 0.1 -0.89 -2.34 0.86
1 1 460 16 124 0.11 2.73 -1.15 -0.24
1 1 461 14 697 0.07 -1.72 1.06 -1.00
1 1 462 7 262 0.04 0.05 -1.15 0.53
1 1 463 11 247 0.06 -0.75 -1.25 0.84
1 1 464 15 457 0.07 1.15 0.15 0.54
1 1 465 8 348 0.07 1.52 -0.50 -0.92
1 1 466 6 19 0.07 -0.64 -2.93 -0.06
1 1 467 12 535 0.08 2.52 0.57 -0.81
1 1 468 13 470 0.08 2.04 0.28 0.99
1 1 469 6 665 0.06 2.11 1.09 -0.92
1 1 470 14 83 0.1 -1.33 -2.34 0.72
1 1 471 5 344 0.04 0.98 -0.67 0.58
1 1 472 6 519 0.05 1.59 0.53 -0.94
1 1 473 12 228 0.08 0.99 -1.19 0.89
1 1 474 6 663 0.04 0.34 1.27 0.72
1 1 475 17 601 0.06 -0.04 1.00 -0.02
1 1 476 10 203 0.06 -1.46 -1.31 -1.00
1 1 477 8 865 0.11 -0.35 2.70 0.69
1 1 478 10 653 0.06 -0.90 1.11 -0.82
1 1 479 7 204 0.08 2.88 -0.59 -0.33
1 1 480 15 340 0.11 -2.69 -0.65 -0.63
1 1 481 15 744 0.08 -1.22 1.56 -1.00
1 1 482 13 822 0.11 -1.88 1.79 -0.80
1 1 483 9 589 0.07 0.56 0.99 -0.51
1 1 484 12 549 0.07 0.70 0.75 0.24
1 1 485 7 110 0.09 0.53 -2.08 0.98
1 1 486 8 291 0.04 0.14 -1.01 -0.20
1 1 487 21 586 0.08 0.26 0.97 0.08
1 1 488 13 666 0.09 1.40 1.26 -0.99
1 1 489 11 349 0.08 -0.92 -0.78 -0.61
1 1 490 9 171 0.08 -1.75 -1.42 -0.97
1 1 491 8 635 0.05 -0.52 1.12 -0.64
1 1 492 11 487 0.06 0.95 0.36 0.17
1 1 493 14 767 0.13 2.54 1.53 -0.23
1 1 494 17 175 0.07 0.64 -1.57 0.95
1 1 495 14 96 0.12 -2.30 -1.76 -0.42
1 1 496 9 428 0.04 -1.03 -0.25 -0.34
1 1 497 10 447 0.05 1.00 0.08 -0.12
1 1 498 10 466 0.1 2.97 0.21 -0.20
1 1 499 10 879 0.13 -1.37 2.46 -0.56
1 1 500 12 806 0.11 -2.61 1.27 -0.42

Now let us understand what each column in the above table means:

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Now, let’s check the compression summary for HVT (torus_mapA) where n_cell was set to 900. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapA_compression_summary <- torus_mapA[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapA_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 900 751 0.83 n_cells: 900 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 83% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

Now let’s try to understand plotHVT function. The parameters have been explained in detail below:

plotHVT <-(hvt.results, line.width, color.vec, pch1 = 21, palette.color = 6, centroid.size = 1.5, title = NULL, maxDepth = NULL, dataset, child.level, hmap.cols, previous_level_heatmap = TRUE, show.points = FALSE, asp = 1, ask = TRUE, tess.label = NULL, quant.error.hmap = NULL, n_cells.hmap = NULL, label.size = 0.5, sepration_width = 7, layer_opacity = c(0.5, 0.75, 0.99), dim_size = 1000, heatmap = '2DHVT') 

Let’s plot the Voronoi tessellation for layer 1 (map A).

plotHVT(torus_mapA,
        line.width = c(0.4), 
        color.vec = c("#141B41"),
        centroid.size = 0.01,
        maxDepth = 1, heatmap = '2DHVT') 
Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 900 cells in the dataset ’computers’

Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 900 cells in the dataset ’computers’

Heat Maps

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data.

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  plotHVT(
  torus_mapA,
  traindata,
  child.level = 1,
  hmap.cols = "x",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 900,
  heatmap = '2DHEATMAP'
) 
Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’x’ in the ’torus’ dataset

Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’x’ in the ’torus’ dataset


  plotHVT(
  torus_mapA,
  traindata,
  child.level = 1,
  hmap.cols = "y",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 900,
  heatmap = '2DHEATMAP'
) 
Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’y’ in the ’torus’ dataset

Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’y’ in the ’torus’ dataset


  plotHVT(
  torus_mapA,
  traindata,
  child.level = 1,
  hmap.cols = "z",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
  heatmap = '2DHEATMAP'
) 
Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’z’ in the ’torus’ dataset

Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’z’ in the ’torus’ dataset

4 Map B : Compressed Novelty Map

Let us try to visualize the Map B from the flow diagram below.

Figure 10: Data Segregation with highlighted bounding box in red around map B

Figure 10: Data Segregation with highlighted bounding box in red around map B

In this section, we will manually figure out the novelty cells from the plotted torus_mapA and store it in identified_Novelty_cells variable.

Note: For manual selecting the novelty cells from map A, one can enhance its interactivity by adding plotly elements to the code. This will transform map A into an interactive plot, allowing users to actively engage with the data. By hovering over the centroids of the cells, a tag containing segment child information will be displayed. Users can explore the map by hovering over different cells and selectively choose the novelty cells they wish to consider. Added an image for reference.

Figure 11: Manually selecting novelty cells

Figure 11: Manually selecting novelty cells

The removeNovelty function removes the identified novelty cell(s) from the training dataset (containing 9600 datapoints) and stores those records separately.

It takes input as the cell number (Segment.Child) of the manually identified novelty cell(s) and the compressed HVT map (torus_mapA) with 900 cells. It returns a list of two items: data with novelty, and a subset of the data without novelty.

NOTE: As we are using torus data here, the identified novelty cells here given are for demo purpose.

identified_Novelty_cells <<- c(69,246,279,293,573,883,706,745)   #as a example
output_list <- removeNovelty(identified_Novelty_cells, torus_mapA)
data_with_novelty <- output_list[[1]]
data_without_novelty <- output_list[[2]]

Let’s have a look at the data with novelty(containing 99 records). For the sake of brevity, we will only show the first 20 rows.

novelty_data <- data_with_novelty
novelty_data$Row.No <- row.names(novelty_data)
novelty_data <- novelty_data %>% dplyr::select("Row.No","Cell.ID","Cell.Number","x","y","z")
colnames(novelty_data) <- c("Row.No","Cell.ID","Segment.Child","x","y","z")
novelty_data %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = TRUE, limit = 20)
Row.No Cell.ID Segment.Child x y z
1 860 69 1.8040 2.3903 -0.1035
2 860 69 1.8377 2.3630 0.1140
3 860 69 1.8568 2.3561 -0.0185
4 860 69 1.7230 2.4524 0.0754
5 860 69 1.8535 2.3453 -0.1458
6 860 69 1.7133 2.4573 0.0938
7 860 69 1.7514 2.4179 0.1694
8 860 69 1.6960 2.4726 0.0574
9 860 69 1.8780 2.3255 0.1472
10 860 69 1.8998 2.3110 0.1290
11 860 69 1.9351 2.2918 -0.0300
12 860 69 1.6999 2.4717 -0.0212
13 860 69 1.9299 2.2848 -0.1352
14 841 246 2.1032 2.1328 0.0955
15 841 246 2.1324 2.1102 -0.0032
16 841 246 2.1935 2.0402 0.0928
17 841 246 1.9725 2.2469 0.1419
18 841 246 2.2502 1.9782 0.0877
19 841 246 2.1590 2.0823 0.0300
20 841 246 2.0956 2.1403 0.0963

4.1 Voronoi Tessellation with highlighted novelty cell

The plotNovelCells function is used to plot the Voronoi tessellation using the compressed HVT map (torus_mapA) containing 900 cells and highlights the identified novelty cell(s) i.e 8 cells (containing 99 records) in red on the map.

plotNovelCells(identified_Novelty_cells, torus_mapA,line.width = c(0.4),centroid.size = 0.01)
Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

We pass the dataframe with novelty records (99 records) to trainHVT function along with other model parameters mentioned below to generate map B (layer2)

Model Parameters

colnames(data_with_novelty) <- c("Cell.ID","Segment.Child","x","y","z")
data_with_novelty <- data_with_novelty[,-1:-2]
torus_mapB <- list()
mapA_scale_summary = torus_mapA[[3]]$scale_summary
torus_mapB <- trainHVT(data_with_novelty,
                  n_cells = 6,   
                  depth = 1,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = FALSE,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = FALSE
                  )

The datatable displayed below is the summary from map B (layer 2) showing Cell.ID, Centroids and Quantization Error for each of the 6 cells.

summaryTable(torus_mapB[[3]]$summary,scroll = TRUE,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error x y z
1 1 1 13 3 0.17 2.46 -1.71 0.08
1 1 2 10 1 0.11 -1.45 -2.62 -0.07
1 1 3 23 5 0.17 2.26 1.95 0.07
1 1 4 12 4 0.09 2.64 -1.40 0.03
1 1 5 14 6 0.13 1.83 2.37 0.03
1 1 6 27 2 0.23 1.99 -2.22 0.05

Now let’s check the compression summary for HVT (torus_mapB). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapB_compression_summary <- torus_mapB[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapB_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 6 5 0.83 n_cells: 6 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 83% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

5 Map C : Compressed Map without Novelty

Let us try to visualize the compressed Map C from the flow diagram below.

Figure 13:Data Segregation with highlighted bounding box in red around compressed map C

Figure 13:Data Segregation with highlighted bounding box in red around compressed map C

5.0.1 Iteration 1:

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the dataset without Novelty (containing 4983 records) and below mentioned model parameters.

Model Parameters

torus_mapC <- list()
mapA_scale_summary = torus_mapA[[3]]$scale_summary
torus_mapC <- trainHVT(data_without_novelty,
                  n_cells = 10,
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = FALSE,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = FALSE,
                  scale_summary = mapA_scale_summary)

Now let’s check the compression summary for HVT (torus_mapC) where n_cell was set to 10. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- torus_mapC[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 10 0 0 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 100 0 0 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 0% of the cells have hit the quantization threshold error in level 2

5.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression at depth 2. Let’s try to compress again using the below mentioned set of model parameters and the dataset without novelty (containing 9501 records).

Model Parameters

torus_mapC <- list()
torus_mapC <- trainHVT(data_without_novelty,
                  n_cells = 20,    
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = FALSE,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = FALSE,
                  scale_summary = mapA_scale_summary)

The datatable displayed below is the summary from map C (layer2). showing Cell.ID, Centroids and Quantization Error for each of the 367 cells.

summaryTable(torus_mapC[[3]]$summary,scroll = TRUE,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error x y z
1 1 1 412 330 0.62 2.22 0.95 -0.69
1 1 2 452 51 0.66 -2.27 0.05 -0.73
1 1 3 404 193 0.64 0.81 -2.51 -0.35
1 1 4 490 144 0.58 -0.21 -1.52 0.78
1 1 5 363 340 0.59 2.67 -0.13 0.41
1 1 6 578 197 0.66 -0.20 1.39 0.70
1 1 7 489 35 0.69 -2.12 -0.97 0.71
1 1 8 587 240 0.65 0.65 1.01 -0.52
1 1 9 394 264 0.71 1.49 -1.51 0.83
1 1 10 431 67 0.74 -1.48 -1.30 -0.85
1 1 11 571 173 0.68 0.10 -1.14 -0.46
1 1 12 566 249 0.58 1.08 -0.12 0.32
1 1 13 482 76 0.64 -2.16 1.18 0.58
1 1 14 504 153 0.64 -1.00 1.42 -0.82
1 1 15 439 307 0.63 1.58 1.35 0.84
1 1 16 422 293 0.68 0.72 2.55 -0.27
1 1 17 488 283 0.72 1.78 -0.93 -0.81
1 1 18 362 46 0.64 -1.04 -2.55 0.06
1 1 19 718 127 0.56 -1.03 0.01 0.17
1 1 20 349 161 0.64 -1.03 2.53 0.23
2 1 1 32 335 0.16 2.42 0.67 -0.85
2 1 2 13 366 0.09 2.73 1.19 -0.16
2 1 3 17 365 0.12 2.81 1.03 0.07
2 1 4 27 317 0.14 1.95 0.83 -0.99
2 1 5 14 360 0.14 2.89 0.55 -0.30
2 1 6 29 296 0.13 1.53 0.87 -0.97
2 1 7 40 338 0.17 2.27 1.22 -0.80
2 1 8 17 306 0.11 1.91 0.23 -0.99
2 1 9 18 367 0.15 2.64 1.39 0.11
2 1 10 23 356 0.2 2.13 1.95 -0.43
2 1 11 21 342 0.17 2.63 0.12 -0.76
2 1 12 31 318 0.17 1.69 1.51 -0.95
2 1 13 32 294 0.16 1.61 0.42 -0.93
2 1 14 16 358 0.15 2.73 0.98 -0.41
2 1 15 18 362 0.16 2.53 1.45 -0.37
2 1 16 16 351 0.13 2.73 0.54 -0.61
2 1 17 27 328 0.14 2.32 0.24 -0.94
2 1 18 21 339 0.2 1.93 1.79 -0.76
2 1 19 0 NA NA NA NA NA
2 1 20 0 NA NA NA NA NA
2 2 1 23 52 0.17 -2.51 1.06 -0.67
2 2 2 13 45 0.13 -2.38 0.13 -0.91
2 2 3 30 58 0.12 -1.92 -0.41 -0.99
2 2 4 17 14 0.13 -2.65 -0.91 -0.57
2 2 5 19 18 0.14 -2.92 0.07 -0.37
2 2 6 16 9 0.18 -2.94 -0.48 -0.13
2 2 7 15 26 0.14 -2.86 0.62 -0.34
2 2 8 31 85 0.14 -1.52 -0.23 -0.88
2 2 9 25 24 0.14 -2.67 -0.25 -0.72
2 2 10 16 74 0.15 -2.11 0.95 -0.94
2 2 11 21 55 0.15 -2.30 0.60 -0.92
2 2 12 26 63 0.11 -2.00 0.14 -1.00
2 2 13 23 17 0.17 -2.98 0.10 0.06
2 2 14 24 31 0.14 -2.67 0.36 -0.70
2 2 15 15 42 0.1 -2.28 -0.27 -0.95
2 2 16 36 89 0.15 -1.73 0.54 -0.97
2 2 17 26 100 0.11 -1.40 0.15 -0.81
2 2 18 28 15 0.12 -2.80 -0.56 -0.50
2 2 19 26 86 0.11 -1.62 0.12 -0.92
2 2 20 22 28 0.12 -2.41 -0.66 -0.86
2 3 1 22 188 0.15 0.77 -2.87 0.17
2 3 2 29 177 0.17 0.52 -2.47 -0.84
2 3 3 33 280 0.17 1.74 -2.24 -0.51
2 3 4 22 160 0.15 0.36 -2.76 0.60
2 3 5 24 170 0.12 0.34 -2.11 -0.98
2 3 6 26 196 0.16 0.82 -2.58 0.69
2 3 7 20 138 0.18 0.23 -2.96 0.18
2 3 8 18 178 0.2 0.58 -2.81 -0.46
2 3 9 27 133 0.18 0.08 -2.85 -0.50
2 3 10 25 132 0.17 -0.02 -2.43 -0.89
2 3 11 36 251 0.19 1.36 -2.60 0.31
2 3 12 27 207 0.15 0.89 -1.94 -0.98
2 3 13 34 214 0.16 1.04 -2.48 -0.71
2 3 14 19 208 0.13 0.98 -2.77 -0.32
2 3 15 23 235 0.12 1.29 -2.07 -0.89
2 3 16 19 252 0.17 1.38 -2.60 -0.25
2 3 17 0 NA NA NA NA NA
2 3 18 0 NA NA NA NA NA
2 3 19 0 NA NA NA NA NA
2 3 20 0 NA NA NA NA NA
2 4 1 49 143 0.16 -0.29 -1.30 0.74
2 4 2 35 163 0.13 -0.08 -1.09 0.41
2 4 3 27 147 0.13 0.06 -2.09 0.99
2 4 4 42 97 0.17 -0.87 -1.34 0.91
2 4 5 33 142 0.12 -0.43 -0.97 0.34
2 4 6 35 187 0.16 0.45 -1.39 0.84
2 4 7 32 110 0.16 -0.40 -1.79 0.98
2 4 8 32 155 0.14 -0.01 -1.62 0.92
2 4 9 38 71 0.17 -1.00 -1.80 0.99
2 4 10 30 145 0.2 0.14 -2.48 0.86
2 4 11 16 96 0.18 -0.41 -2.31 0.92
2 4 12 40 183 0.17 0.44 -1.80 0.98
2 4 13 43 119 0.18 -0.71 -1.00 0.63
2 4 14 38 184 0.17 0.31 -1.16 0.60
2 4 15 0 NA NA NA NA NA
2 4 16 0 NA NA NA NA NA
2 4 17 0 NA NA NA NA NA
2 4 18 0 NA NA NA NA NA
2 4 19 0 NA NA NA NA NA
2 4 20 0 NA NA NA NA NA
2 5 1 18 312 0.12 1.98 0.01 0.99
2 5 2 17 361 0.16 2.78 0.85 0.41
2 5 3 19 325 0.17 2.40 -0.72 0.85
2 5 4 22 333 0.11 2.48 -0.17 0.86
2 5 5 18 314 0.11 2.13 -0.43 0.98
2 5 6 22 346 0.15 2.84 -0.74 0.31
2 5 7 16 344 0.15 2.73 -0.49 0.63
2 5 8 27 347 0.18 2.87 -0.20 -0.45
2 5 9 25 354 0.15 2.94 -0.12 0.30
2 5 10 15 357 0.14 2.96 0.19 -0.21
2 5 11 21 329 0.14 2.21 0.34 0.96
2 5 12 14 364 0.11 2.93 0.61 0.05
2 5 13 23 341 0.17 2.83 -0.89 -0.19
2 5 14 25 348 0.15 2.67 0.19 0.72
2 5 15 24 359 0.12 2.91 0.37 0.33
2 5 16 19 334 0.16 2.70 -1.13 0.33
2 5 17 20 349 0.13 2.56 0.61 0.77
2 5 18 18 352 0.15 2.96 -0.41 0.01
2 5 19 0 NA NA NA NA NA
2 5 20 0 NA NA NA NA NA
2 6 1 38 216 0.16 -0.05 1.70 0.95
2 6 2 41 182 0.15 -0.39 0.92 0.05
2 6 3 46 238 0.17 0.52 1.11 0.62
2 6 4 40 254 0.14 0.46 1.56 0.92
2 6 5 39 218 0.16 0.12 1.30 0.71
2 6 6 35 171 0.12 -0.78 1.76 0.99
2 6 7 38 175 0.16 -0.57 0.93 0.43
2 6 8 36 203 0.16 -0.33 2.10 0.98
2 6 9 41 150 0.15 -1.04 1.38 0.95
2 6 10 30 195 0.1 -0.12 1.00 0.14
2 6 11 40 157 0.17 -0.87 1.03 0.75
2 6 12 36 213 0.14 0.15 1.05 0.35
2 6 13 28 268 0.14 0.47 2.05 0.98
2 6 14 34 185 0.15 -0.45 1.51 0.90
2 6 15 34 191 0.14 -0.24 1.17 0.59
2 6 16 22 245 0.15 0.14 2.29 0.94
2 6 17 0 NA NA NA NA NA
2 6 18 0 NA NA NA NA NA
2 6 19 0 NA NA NA NA NA
2 6 20 0 NA NA NA NA NA
2 7 1 31 2 0.21 -2.60 -1.47 -0.02
2 7 2 20 16 0.14 -2.82 -0.20 0.54
2 7 3 24 30 0.17 -2.47 -0.12 0.87
2 7 4 17 36 0.13 -2.18 -0.57 0.96
2 7 5 22 4 0.15 -2.81 -1.00 0.06
2 7 6 18 1 0.16 -2.25 -1.94 0.17
2 7 7 28 49 0.12 -1.78 -0.95 0.99
2 7 8 20 8 0.14 -2.86 -0.60 0.36
2 7 9 25 6 0.18 -2.55 -1.29 0.49
2 7 10 33 57 0.12 -1.47 -1.34 0.99
2 7 11 20 33 0.13 -1.64 -1.75 0.91
2 7 12 22 20 0.12 -2.55 -0.47 0.80
2 7 13 23 77 0.12 -1.52 -0.60 0.92
2 7 14 22 54 0.11 -2.01 -0.05 0.99
2 7 15 36 59 0.16 -1.84 -0.41 0.99
2 7 16 18 5 0.13 -2.29 -1.71 0.50
2 7 17 34 22 0.15 -2.06 -1.41 0.85
2 7 18 13 11 0.15 -1.96 -1.94 0.64
2 7 19 32 82 0.12 -1.25 -0.98 0.91
2 7 20 31 19 0.13 -2.34 -0.99 0.83
2 8 1 29 219 0.1 0.27 0.97 0.05
2 8 2 35 242 0.12 0.70 0.91 -0.52
2 8 3 35 237 0.12 0.69 0.77 -0.23
2 8 4 30 253 0.13 0.94 0.48 -0.34
2 8 5 28 212 0.14 0.11 1.13 -0.50
2 8 6 18 241 0.1 0.79 0.61 0.02
2 8 7 36 199 0.15 -0.07 1.02 -0.22
2 8 8 48 226 0.13 0.39 1.13 -0.59
2 8 9 35 257 0.14 0.62 1.45 -0.90
2 8 10 18 239 0.14 0.33 1.76 -0.97
2 8 11 24 217 0.12 0.11 1.39 -0.79
2 8 12 23 289 0.16 1.24 1.16 -0.94
2 8 13 39 287 0.15 0.99 1.63 -0.99
2 8 14 33 266 0.14 1.21 0.38 -0.67
2 8 15 28 224 0.12 0.43 0.93 -0.23
2 8 16 25 261 0.1 0.98 0.70 -0.61
2 8 17 29 274 0.11 1.25 0.78 -0.85
2 8 18 36 230 0.14 0.58 0.84 0.21
2 8 19 38 263 0.14 0.88 1.14 -0.82
2 8 20 0 NA NA NA NA NA
2 9 1 38 247 0.14 1.27 -1.35 0.98
2 9 2 19 321 0.2 2.37 -1.76 0.27
2 9 3 29 206 0.15 0.77 -1.56 0.95
2 9 4 31 275 0.17 1.67 -1.81 0.88
2 9 5 29 276 0.13 1.64 -1.12 0.99
2 9 6 30 305 0.17 2.11 -1.20 0.89
2 9 7 36 288 0.17 1.66 -0.77 0.97
2 9 8 48 220 0.19 0.92 -1.06 0.80
2 9 9 26 303 0.15 2.04 -1.87 0.62
2 9 10 20 319 0.2 2.35 -1.49 0.61
2 9 11 30 225 0.13 1.10 -1.83 0.98
2 9 12 15 232 0.14 1.24 -2.30 0.77
2 9 13 22 202 0.17 0.79 -2.19 0.93
2 9 14 21 272 0.16 1.63 -2.30 0.55
2 9 15 0 NA NA NA NA NA
2 9 16 0 NA NA NA NA NA
2 9 17 0 NA NA NA NA NA
2 9 18 0 NA NA NA NA NA
2 9 19 0 NA NA NA NA NA
2 9 20 0 NA NA NA NA NA
2 10 1 32 75 0.12 -1.34 -1.19 -0.97
2 10 2 21 95 0.11 -1.15 -0.87 -0.83
2 10 3 25 69 0.13 -1.56 -0.93 -0.97
2 10 4 20 48 0.11 -1.91 -0.85 -0.99
2 10 5 16 111 0.12 -0.86 -0.85 -0.61
2 10 6 21 7 0.18 -2.08 -1.97 -0.46
2 10 7 11 90 0.09 -0.62 -1.92 -0.99
2 10 8 26 10 0.19 -2.29 -1.65 -0.54
2 10 9 31 62 0.14 -1.31 -1.62 -0.99
2 10 10 23 106 0.1 -1.10 -0.62 -0.67
2 10 11 31 40 0.17 -1.87 -1.28 -0.95
2 10 12 17 88 0.1 -1.37 -0.54 -0.85
2 10 13 14 70 0.09 -1.66 -0.58 -0.97
2 10 14 31 104 0.13 -0.86 -1.20 -0.85
2 10 15 25 29 0.16 -1.79 -1.79 -0.83
2 10 16 29 21 0.17 -2.30 -1.13 -0.81
2 10 17 14 39 0.18 -1.41 -2.08 -0.85
2 10 18 22 68 0.12 -0.99 -1.99 -0.97
2 10 19 22 93 0.13 -0.82 -1.53 -0.96
2 10 20 0 NA NA NA NA NA
2 11 1 56 162 0.17 -0.08 -1.08 -0.40
2 11 2 75 186 0.18 0.34 -0.94 0.11
2 11 3 46 190 0.16 0.44 -0.96 -0.34
2 11 4 33 205 0.12 0.69 -1.09 -0.70
2 11 5 65 151 0.18 -0.31 -0.94 0.01
2 11 6 27 118 0.15 -0.32 -1.80 -0.97
2 11 7 31 164 0.16 0.17 -1.75 -0.96
2 11 8 32 201 0.15 0.71 -1.47 -0.92
2 11 9 39 154 0.16 -0.07 -1.37 -0.77
2 11 10 41 180 0.15 0.31 -1.25 -0.70
2 11 11 41 211 0.13 0.75 -0.83 -0.46
2 11 12 56 136 0.17 -0.58 -0.91 -0.39
2 11 13 29 125 0.16 -0.49 -1.29 -0.77
2 11 14 0 NA NA NA NA NA
2 11 15 0 NA NA NA NA NA
2 11 16 0 NA NA NA NA NA
2 11 17 0 NA NA NA NA NA
2 11 18 0 NA NA NA NA NA
2 11 19 0 NA NA NA NA NA
2 11 20 0 NA NA NA NA NA
2 12 1 60 260 0.2 1.13 0.11 0.49
2 12 2 45 278 0.17 1.50 -0.37 0.88
2 12 3 51 256 0.13 0.94 0.57 0.43
2 12 4 40 246 0.14 1.02 0.01 -0.21
2 12 5 29 273 0.15 1.28 0.38 0.74
2 12 6 34 221 0.11 0.87 -0.58 0.29
2 12 7 32 210 0.13 0.73 -0.86 0.49
2 12 8 19 233 0.13 1.02 -0.38 -0.41
2 12 9 27 222 0.13 0.87 -0.53 -0.19
2 12 10 54 244 0.18 1.12 -0.51 0.64
2 12 11 31 255 0.17 1.15 -0.05 -0.54
2 12 12 37 292 0.2 1.56 0.07 0.89
2 12 13 34 236 0.13 0.99 -0.25 0.20
2 12 14 18 215 0.11 0.74 -0.68 0.03
2 12 15 55 250 0.15 0.95 0.31 0.07
2 12 16 0 NA NA NA NA NA
2 12 17 0 NA NA NA NA NA
2 12 18 0 NA NA NA NA NA
2 12 19 0 NA NA NA NA NA
2 12 20 0 NA NA NA NA NA
2 13 1 12 123 0.08 -1.37 1.31 0.99
2 13 2 24 81 0.16 -2.01 1.91 0.61
2 13 3 19 64 0.17 -2.30 1.53 0.62
2 13 4 36 61 0.22 -2.36 1.67 -0.42
2 13 5 24 25 0.19 -2.82 0.53 0.45
2 13 6 27 50 0.17 -2.38 0.61 0.88
2 13 7 34 73 0.13 -2.04 0.78 0.97
2 13 8 24 43 0.14 -2.69 1.11 0.39
2 13 9 20 120 0.13 -1.33 0.93 0.92
2 13 10 25 102 0.17 -1.68 1.78 0.88
2 13 11 26 105 0.12 -1.60 1.10 0.99
2 13 12 23 80 0.13 -2.00 1.19 0.93
2 13 13 23 103 0.11 -1.67 1.50 0.96
2 13 14 20 56 0.15 -2.42 1.06 0.76
2 13 15 20 32 0.22 -2.62 0.28 0.76
2 13 16 32 47 0.18 -2.56 1.54 0.08
2 13 17 16 60 0.14 -2.10 0.37 0.98
2 13 18 29 66 0.17 -2.21 1.99 0.09
2 13 19 26 38 0.25 -2.77 1.05 -0.18
2 13 20 22 94 0.13 -1.62 0.57 0.96
2 14 1 24 165 0.13 -0.73 0.84 -0.46
2 14 2 33 200 0.15 -0.37 2.14 -0.97
2 14 3 31 92 0.21 -1.84 1.93 -0.72
2 14 4 27 117 0.14 -1.39 0.85 -0.93
2 14 5 34 148 0.12 -1.09 1.50 -0.98
2 14 6 21 204 0.14 -0.18 1.67 -0.94
2 14 7 33 169 0.14 -0.71 1.19 -0.78
2 14 8 26 98 0.11 -1.69 1.11 -0.99
2 14 9 6 172 0.06 -0.58 0.84 -0.20
2 14 10 19 122 0.13 -1.25 0.61 -0.79
2 14 11 26 176 0.11 -0.83 2.29 -0.89
2 14 12 27 189 0.16 -0.32 1.22 -0.67
2 14 13 25 139 0.22 -1.30 2.07 -0.88
2 14 14 25 146 0.18 -1.04 1.05 -0.85
2 14 15 25 114 0.13 -1.49 1.51 -0.98
2 14 16 27 79 0.19 -2.08 1.46 -0.82
2 14 17 32 179 0.13 -0.54 1.54 -0.92
2 14 18 22 181 0.14 -0.44 0.97 -0.36
2 14 19 20 149 0.12 -0.94 0.76 -0.61
2 14 20 21 168 0.11 -0.83 1.81 -0.99
2 15 1 21 337 0.17 1.56 2.25 0.65
2 15 2 19 309 0.15 1.13 2.20 0.87
2 15 3 33 302 0.12 1.45 1.18 0.99
2 15 4 34 290 0.16 0.91 1.76 0.99
2 15 5 15 363 0.14 2.45 1.49 0.46
2 15 6 27 326 0.14 1.87 1.34 0.94
2 15 7 17 353 0.13 2.53 1.14 0.62
2 15 8 30 286 0.13 1.23 0.89 0.87
2 15 9 20 350 0.15 2.13 1.65 0.71
2 15 10 31 262 0.14 0.86 0.90 0.66
2 15 11 20 323 0.13 2.03 0.59 0.98
2 15 12 25 336 0.13 2.23 0.93 0.90
2 15 13 39 282 0.18 1.01 1.30 0.93
2 15 14 26 355 0.2 2.03 2.02 0.46
2 15 15 30 315 0.1 1.79 0.91 0.99
2 15 16 17 295 0.14 1.52 0.61 0.92
2 15 17 35 320 0.17 1.56 1.77 0.92
2 15 18 0 NA NA NA NA NA
2 15 19 0 NA NA NA NA NA
2 15 20 0 NA NA NA NA NA
2 16 1 23 297 0.15 0.92 2.18 -0.92
2 16 2 38 316 0.25 1.02 2.61 0.56
2 16 3 20 308 0.14 0.86 2.84 -0.22
2 16 4 35 285 0.2 0.44 2.62 0.74
2 16 5 14 299 0.15 0.55 2.93 0.16
2 16 6 19 304 0.14 1.24 2.00 -0.93
2 16 7 16 234 0.1 0.16 2.02 -1.00
2 16 8 18 343 0.15 1.53 2.51 0.28
2 16 9 17 228 0.1 -0.01 2.36 -0.93
2 16 10 26 331 0.2 1.19 2.73 -0.07
2 16 11 25 345 0.19 1.71 2.36 -0.37
2 16 12 20 227 0.15 -0.13 2.63 -0.77
2 16 13 21 271 0.14 0.24 2.86 0.47
2 16 14 12 243 0.12 -0.10 2.88 -0.45
2 16 15 27 300 0.16 0.75 2.62 -0.67
2 16 16 27 324 0.16 1.33 2.39 -0.67
2 16 17 15 279 0.15 0.29 2.93 -0.31
2 16 18 18 265 0.15 0.24 2.63 -0.75
2 16 19 19 270 0.11 0.52 2.20 -0.96
2 16 20 12 269 0.1 0.10 2.99 0.00
2 17 1 29 311 0.16 2.35 -1.00 -0.81
2 17 2 20 258 0.12 1.37 -1.05 -0.96
2 17 3 26 332 0.16 2.62 -1.27 -0.38
2 17 4 40 281 0.17 1.54 -0.07 -0.87
2 17 5 21 291 0.16 1.86 -1.92 -0.72
2 17 6 18 310 0.18 2.27 -1.88 -0.30
2 17 7 37 301 0.16 1.98 -0.51 -0.99
2 17 8 15 313 0.16 2.34 -1.61 -0.53
2 17 9 22 229 0.12 1.13 -1.21 -0.93
2 17 10 20 298 0.16 2.04 -1.46 -0.85
2 17 11 24 322 0.2 2.36 -0.26 -0.91
2 17 12 34 248 0.19 1.34 -1.60 -0.98
2 17 13 25 327 0.17 2.58 -0.75 -0.70
2 17 14 26 284 0.11 1.72 -0.87 -0.99
2 17 15 29 267 0.15 1.47 -0.61 -0.91
2 17 16 35 259 0.15 1.29 -0.34 -0.74
2 17 17 38 231 0.13 1.07 -0.72 -0.70
2 17 18 29 277 0.13 1.74 -1.31 -0.98
2 17 19 0 NA NA NA NA NA
2 17 20 0 NA NA NA NA NA
2 18 1 30 78 0.15 -0.33 -2.97 0.03
2 18 2 18 41 0.17 -1.19 -2.39 -0.73
2 18 3 23 72 0.17 -0.70 -2.39 -0.85
2 18 4 30 65 0.19 -0.73 -2.47 0.80
2 18 5 22 13 0.15 -1.70 -2.39 -0.33
2 18 6 33 84 0.2 -0.29 -2.82 0.52
2 18 7 13 87 0.18 -0.34 -2.70 -0.67
2 18 8 22 3 0.14 -2.04 -2.18 0.00
2 18 9 28 23 0.17 -1.31 -2.57 0.43
2 18 10 20 44 0.13 -0.88 -2.81 0.30
2 18 11 30 34 0.17 -1.36 -2.15 0.82
2 18 12 19 37 0.16 -0.97 -2.82 -0.14
2 18 13 22 53 0.17 -0.74 -2.75 -0.51
2 18 14 27 12 0.18 -1.75 -2.29 0.43
2 18 15 25 27 0.15 -1.28 -2.60 -0.42
2 18 16 0 NA NA NA NA NA
2 18 17 0 NA NA NA NA NA
2 18 18 0 NA NA NA NA NA
2 18 19 0 NA NA NA NA NA
2 18 20 0 NA NA NA NA NA
2 19 1 44 116 0.14 -1.06 -0.27 0.42
2 19 2 45 137 0.17 -1.07 0.60 0.63
2 19 3 26 115 0.13 -0.96 -0.51 -0.41
2 19 4 28 129 0.13 -0.76 -0.66 -0.07
2 19 5 46 131 0.11 -1.04 0.10 0.29
2 19 6 38 140 0.13 -0.95 0.31 -0.09
2 19 7 30 99 0.14 -1.18 -0.53 0.71
2 19 8 37 141 0.15 -0.95 0.41 0.26
2 19 9 29 126 0.09 -0.99 -0.13 0.05
2 19 10 36 124 0.11 -0.91 -0.42 -0.02
2 19 11 55 130 0.18 -1.10 0.29 -0.50
2 19 12 36 101 0.24 -1.44 0.32 0.84
2 19 13 36 91 0.19 -1.42 -0.18 0.82
2 19 14 39 159 0.18 -0.76 0.69 0.21
2 19 15 34 108 0.16 -1.20 -0.15 -0.62
2 19 16 40 112 0.12 -1.20 0.08 0.60
2 19 17 23 113 0.13 -0.91 -0.71 0.52
2 19 18 31 134 0.1 -0.72 -0.73 0.23
2 19 19 30 121 0.13 -1.04 -0.15 -0.32
2 19 20 35 156 0.15 -0.80 0.63 -0.21
2 20 1 16 194 0.15 -0.49 2.46 0.85
2 20 2 27 166 0.16 -1.01 2.57 -0.62
2 20 3 25 192 0.17 -0.67 2.90 -0.08
2 20 4 27 152 0.18 -1.19 2.72 -0.14
2 20 5 24 109 0.16 -1.54 2.26 0.66
2 20 6 25 198 0.16 -0.55 2.73 -0.61
2 20 7 18 83 0.16 -1.93 2.26 0.15
2 20 8 29 174 0.15 -0.92 2.71 0.49
2 20 9 20 135 0.09 -1.33 1.95 0.93
2 20 10 22 223 0.17 -0.17 2.66 0.73
2 20 11 28 107 0.19 -1.69 2.36 -0.39
2 20 12 16 158 0.09 -1.01 2.12 0.94
2 20 13 21 167 0.14 -0.95 2.40 0.80
2 20 14 30 128 0.17 -1.42 2.56 0.35
2 20 15 21 209 0.18 -0.42 2.91 0.30
2 20 16 0 NA NA NA NA NA
2 20 17 0 NA NA NA NA NA
2 20 18 0 NA NA NA NA NA
2 20 19 0 NA NA NA NA NA
2 20 20 0 NA NA NA NA NA

Now let’s check the compression summary for HVT (torus_mapC). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- torus_mapC[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 20 0 0 n_cells: 20 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 347 338 0.97 n_cells: 20 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 97% of the cells have hit the quantization threshold error in level 2

Let’s plot the Voronoi tessellation for layer 2 (map C)

plotHVT(torus_mapC,
        line.width = c(0.4,0.2), 
        color.vec = c("#141B41","#0582CA"),
        centroid.size = 0.1,
        maxDepth = 2, 
        heatmap = '2DHVT') 
Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 347 cells in the dataset ’torus’ at level 2

Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 347 cells in the dataset ’torus’ at level 2

Heat Maps

Now let’s plot all the features for each cell at level two as a heatmap for better visualization.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (x,y,z). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  plotHVT(
  torus_mapC,
  traindata,
  child.level = 2,
  hmap.cols = "x",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
  heatmap = '2DHEATMAP'
) 
Figure 15: The Voronoi Tessellation with the heat map overlaid for feature x in the ’torus’ dataset

Figure 15: The Voronoi Tessellation with the heat map overlaid for feature x in the ’torus’ dataset


  plotHVT(
  torus_mapC,
  traindata,
  child.level = 2,
  hmap.cols = "y",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
  heatmap = '2DHEATMAP'
) 
Figure 16: The Voronoi Tessellation with the heat map overlaid for features y in the ’torus’ dataset

Figure 16: The Voronoi Tessellation with the heat map overlaid for features y in the ’torus’ dataset


  plotHVT(
  torus_mapC,
  traindata,
  child.level = 2,
  hmap.cols = "z",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
  heatmap = '2DHEATMAP'
) 
Figure 17: The Voronoi Tessellation with the heat map overlaid for features z in the ’torus’ dataset

Figure 17: The Voronoi Tessellation with the heat map overlaid for features z in the ’torus’ dataset

We now have the set of maps (map A, map B & map C) which will be used to score, which map and cell each test record is assigned to, but before that lets view our test dataset

6 Scoring on Test Data

Now once we have built the model, let us try to score using our test dataset (containing 2400 data points) which cell and which layer each point belongs to.

Testing Dataset

The testing dataset includes the following columns:

Let’s have a look at our randomly selected test dataset containing 2400 datapoints.

Table(head(testdata))
x y z
-1.0308 1.1066 -0.8731
1.8847 0.1895 0.9944
-1.0046 -1.8170 -0.9971
-2.4446 -1.6528 0.3097
-0.3961 -2.1775 0.9770
-1.1130 -0.6516 -0.7040

The scoreLayeredHVT function is used to score the test data using the scored set of maps. This function takes an input - a test data and a set of maps (map A, map B, map C).

Now, Let us understand the scoreLayeredHVT function.

scoreLayeredHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = TRUE, 
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1, 
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of scoreLayeredHVT function has been explained below:

The function predicts based on the HVT maps - map A, map B and map C, constructed using trainHVT function. For each test record, the function will assign that record to Layer1 or Layer2. Layer1 contains the cell ids from map A and Layer 2 contains cell ids from map B (novelty map) and map C (map without novelty).

Scoring Algorithm

The prediction algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the scoring method for a single point in the test dataset:

  1. Calculate the distance between the point and the centroid of all the cells in the first level.
  2. Find the cell whose centroid has minimum distance to the point.
  3. Check if the cell drills down further to form more cells.
  4. If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further.

Note : The Scoring algorithm will not work if some of the variables used to perform quantization are missing. In the test dataset, we should not remove any features


validation_data <- testdata
new_predict <- scoreLayeredHVT(
    data=validation_data,
    torus_mapA,
    torus_mapB,
    torus_mapC,
    normalize = TRUE
  )

Let’s see which cell and layer each point belongs to and check the Mean Absolute Difference for each of the 2400 records.


act_pred <- new_predict[["actual_predictedTable"]]
rownames(act_pred) <- NULL
act_pred %>% head(1000) %>%as.data.frame() %>%Table(scroll = TRUE)
Row.Number act_x act_y act_z Layer1.Cell.ID Layer2.Cell.ID pred_x pred_y pred_z diff
1 -1.0308 1.1066 -0.8731 A650 C153 -1.0030224 1.4214296 -0.8194117 0.1320985
2 1.8847 0.1895 0.9944 A470 C307 1.5763323 1.3461567 0.8384533 0.5403237
3 -1.0046 -1.8170 -0.9971 A138 C67 -1.4757775 -1.2955060 -0.8486053 0.3803887
4 -2.4446 -1.6528 0.3097 A94 C35 -2.1242339 -0.9715258 0.7096658 0.4672020
5 -0.3961 -2.1775 0.9770 A112 C144 -0.2145520 -1.5164273 0.7835631 0.3453525
6 -1.1130 -0.6516 -0.7040 A373 C67 -1.4757775 -1.2955060 -0.8486053 0.3837630
7 0.7520 -2.6044 0.7034 A56 C193 0.8084450 -2.5111158 -0.3496015 0.4009102
8 2.0288 1.9519 0.5790 A802 C307 1.5763323 1.3461567 0.8384533 0.4392214
9 0.7930 2.4428 0.8228 A831 C293 0.7212818 2.5491813 -0.2728450 0.4245815
10 -0.0373 -2.8690 -0.4944 A25 C193 0.8084450 -2.5111158 -0.3496015 0.4494759
11 1.6664 -2.1227 -0.7155 A65 C283 1.7808158 -0.9337703 -0.8105779 0.4661411
12 -0.2545 -1.6160 -0.9314 A163 C173 0.0988042 -1.1378972 -0.4550217 0.4359284
13 2.3377 -0.4287 -0.9263 A290 C283 1.7808158 -0.9337703 -0.8105779 0.3925589
14 1.4772 -0.5194 -0.9008 A348 C283 1.7808158 -0.9337703 -0.8105779 0.2694027
15 -0.4677 1.6539 0.9596 A756 C197 -0.1968420 1.3899623 0.6958189 0.2661923
16 1.1034 -0.7534 -0.7478 A294 C283 1.7808158 -0.9337703 -0.8105779 0.3068546
17 2.0890 -0.6341 -0.9831 A231 C283 1.7808158 -0.9337703 -0.8105779 0.2601255
18 -1.6832 -0.2035 0.9525 A464 C35 -2.1242339 -0.9715258 0.7096658 0.4839646
19 -0.3870 1.0347 0.4455 A622 C197 -0.1968420 1.3899623 0.6958189 0.2652464
20 -0.8512 2.8363 -0.2755 A894 C161 -1.0348261 2.5280129 0.2324966 0.3333032
21 2.2895 -0.4637 0.9419 A300 C340 2.6666782 -0.1317501 0.4147766 0.4120838
22 -2.0314 -0.2369 -0.9990 A422 C51 -2.2655670 0.0495469 -0.7279562 0.2638859
23 -0.5293 -0.8566 0.1173 A341 C127 -1.0331492 0.0055723 0.1700299 0.4729171
24 -1.8571 1.6274 -0.8830 A822 C153 -1.0030224 1.4214296 -0.8194117 0.3745454
25 -0.8999 0.4451 0.0885 A533 C127 -1.0331492 0.0055723 0.1700299 0.2181023
26 0.7219 2.8642 0.3005 A872 C293 0.7212818 2.5491813 -0.2728450 0.2963273
27 0.7150 -0.7791 -0.3341 A328 C173 0.0988042 -1.1378972 -0.4550217 0.3653049
28 -2.9876 -0.2310 0.0831 A454 C51 -2.2655670 0.0495469 -0.7279562 0.6045454
29 2.9383 0.6049 0.0126 A578 C340 2.6666782 -0.1317501 0.4147766 0.4701495
30 2.5807 -1.3975 -0.3553 A85 C283 1.7808158 -0.9337703 -0.8105779 0.5729639
31 1.6829 -2.0223 0.7758 A87 C264 1.4868170 -1.5149655 0.8303546 0.2526574
32 -0.3810 1.7125 -0.9694 A737 C153 -1.0030224 1.4214296 -0.8194117 0.3543604
33 1.0605 1.2803 0.9413 A660 C307 1.5763323 1.3461567 0.8384533 0.2281786
34 0.2569 1.4850 0.8701 A703 C197 -0.1968420 1.3899623 0.6958189 0.2410203
35 1.1535 1.3623 -0.9766 A655 C240 0.6514734 1.0103327 -0.5200428 0.4368504
36 0.5354 1.8832 0.9991 A753 C197 -0.1968420 1.3899623 0.6958189 0.5095870
37 -1.5746 -0.0465 0.9053 A464 C127 -1.0331492 0.0055723 0.1700299 0.4429311
38 -0.9720 -1.2693 0.9160 A223 C144 -0.2145520 -1.5164273 0.7835631 0.3790041
39 -1.5215 1.8082 0.9317 A809 C76 -2.1614253 1.1751861 0.5762133 0.5428086
40 -0.7437 0.8610 0.5065 A584 C197 -0.1968420 1.3899623 0.6958189 0.4217130
41 0.4282 -1.0374 0.4791 A297 C173 0.0988042 -1.1378972 -0.4550217 0.4546716
42 2.4955 0.0918 0.8676 A458 C340 2.6666782 -0.1317501 0.4147766 0.2825173
43 -0.7473 -2.7938 0.4520 A39 C46 -1.0353834 -2.5511392 0.0634906 0.3064179
44 -0.8596 1.5771 -0.9790 A741 C153 -1.0030224 1.4214296 -0.8194117 0.1528937
45 1.0781 -2.6842 0.4509 A16 C193 0.8084450 -2.5111158 -0.3496015 0.4144135
46 0.7559 1.9550 0.9954 A765 C307 1.5763323 1.3461567 0.8384533 0.5287408
47 0.5300 1.3668 0.8455 A681 C197 -0.1968420 1.3899623 0.6958189 0.2998952
48 -2.9305 0.5814 0.1566 A683 C76 -2.1614253 1.1751861 0.5762133 0.5941580
49 2.3888 -1.0670 0.7875 A149 C264 1.4868170 -1.5149655 0.8303546 0.4642677
50 -0.6055 0.8773 -0.3572 A598 C153 -1.0030224 1.4214296 -0.8194117 0.4679546
51 0.7468 -1.9001 0.9991 A140 C264 1.4868170 -1.5149655 0.8303546 0.4312990
52 1.1944 -1.3305 -0.9773 A192 C283 1.7808158 -0.9337703 -0.8105779 0.3832892
53 0.9147 1.2969 -0.9108 A655 C240 0.6514734 1.0103327 -0.5200428 0.3135170
54 1.1413 1.0563 -0.8956 A602 C240 0.6514734 1.0103327 -0.5200428 0.3037837
55 -0.8597 -0.8383 0.6010 A355 C127 -1.0331492 0.0055723 0.1700299 0.4827638
56 1.4374 1.2279 0.9940 A714 C307 1.5763323 1.3461567 0.8384533 0.1375786
57 -1.5625 1.0275 0.9915 A708 C76 -2.1614253 1.1751861 0.5762133 0.3872994
58 -0.3022 -1.3611 0.7956 A216 C144 -0.2145520 -1.5164273 0.7835631 0.0850041
59 -0.7561 -2.5384 -0.7611 A71 C46 -1.0353834 -2.5511392 0.0634906 0.3722044
60 -1.6457 -2.1635 -0.6958 A93 C67 -1.4757775 -1.2955060 -0.8486053 0.3969073
61 -0.7163 -2.3849 -0.8716 A71 C46 -1.0353834 -2.5511392 0.0634906 0.4734711
62 -1.8574 -0.2955 0.9929 A427 C35 -2.1242339 -0.9715258 0.7096658 0.4086980
63 0.5359 1.3339 -0.8268 A656 C240 0.6514734 1.0103327 -0.5200428 0.2486327
64 0.2274 1.1488 0.5593 A640 C197 -0.1968420 1.3899623 0.6958189 0.2673077
65 1.0874 -0.1278 0.4251 A433 C249 1.0778184 -0.1223857 0.3159433 0.0413842
66 1.1080 2.7875 -0.0260 A877 C293 0.7212818 2.5491813 -0.2728450 0.2906273
67 -0.4462 2.1556 -0.9795 A795 C153 -1.0030224 1.4214296 -0.8194117 0.4836937
68 1.4935 2.5288 0.3496 A861 C293 0.7212818 2.5491813 -0.2728450 0.4716815
69 -2.2003 -1.7403 0.5929 A128 C35 -2.1242339 -0.9715258 0.7096658 0.3205354
70 0.3254 1.9173 0.9985 A764 C197 -0.1968420 1.3899623 0.6958189 0.4507536
71 -0.6653 -1.5250 0.9418 A207 C144 -0.2145520 -1.5164273 0.7835631 0.2058525
72 0.5849 -0.8531 -0.2598 A328 C173 0.0988042 -1.1378972 -0.4550217 0.3220382
73 -0.9395 2.6663 -0.5623 A880 C161 -1.0348261 2.5280129 0.2324966 0.3428032
74 -1.6866 2.1137 0.7101 A858 C161 -1.0348261 2.5280129 0.2324966 0.5145634
75 0.2761 -1.1245 0.5393 A258 C144 -0.2145520 -1.5164273 0.7835631 0.3756141
76 1.1576 1.0473 0.8985 A579 C307 1.5763323 1.3461567 0.8384533 0.2592119
77 1.0850 -1.6815 1.0000 A133 C264 1.4868170 -1.5149655 0.8303546 0.2459990
78 2.6678 -0.0291 -0.7442 A420 C340 2.6666782 -0.1317501 0.4147766 0.4209162
79 -0.5183 1.0593 -0.5713 A635 C153 -1.0030224 1.4214296 -0.8194117 0.3649879
80 -0.4761 -2.5549 -0.8008 A60 C46 -1.0353834 -2.5511392 0.0634906 0.4757783
81 1.0102 -1.3455 0.9483 A186 C264 1.4868170 -1.5149655 0.8303546 0.2546760
82 -2.1061 1.9541 0.4877 A857 C76 -2.1614253 1.1751861 0.5762133 0.3075842
83 0.5324 -0.8526 0.1016 A325 C173 0.0988042 -1.1378972 -0.4550217 0.4251716
84 -1.2963 -0.0971 0.7140 A452 C127 -1.0331492 0.0055723 0.1700299 0.3032644
85 1.0031 0.3850 -0.3786 A483 C240 0.6514734 1.0103327 -0.5200428 0.3728007
86 -0.7562 0.7889 -0.4207 A580 C153 -1.0030224 1.4214296 -0.8194117 0.4260212
87 -0.4141 1.6829 -0.9637 A737 C153 -1.0030224 1.4214296 -0.8194117 0.3315604
88 -2.4210 -1.3252 -0.6499 A131 C67 -1.4757775 -1.2955060 -0.8486053 0.3912073
89 -1.6854 -0.3662 -0.9614 A425 C51 -2.2655670 0.0495469 -0.7279562 0.4097859
90 1.2558 -2.2969 -0.7863 A81 C193 0.8084450 -2.5111158 -0.3496015 0.3660898
91 -2.6405 -0.8162 -0.6455 A340 C51 -2.2655670 0.0495469 -0.7279562 0.4410454
92 0.8325 -0.9413 0.6689 A312 C264 1.4868170 -1.5149655 0.8303546 0.4631457
93 1.4609 -0.1864 0.8497 A431 C249 1.0778184 -0.1223857 0.3159433 0.3269509
94 -0.3355 0.9636 0.2005 A595 C197 -0.1968420 1.3899623 0.6958189 0.3534464
95 0.0714 1.3707 0.7787 A678 C197 -0.1968420 1.3899623 0.6958189 0.1234618
96 -1.0089 -0.6007 0.5639 A395 C127 -1.0331492 0.0055723 0.1700299 0.3414638
97 1.0072 -0.3094 -0.3233 A405 C249 1.0778184 -0.1223857 0.3159433 0.2989587
98 1.0167 0.2707 -0.3188 A483 C249 1.0778184 -0.1223857 0.3159433 0.3629824
99 0.5539 -0.8888 0.3037 A320 C249 1.0778184 -0.1223857 0.3159433 0.4341920
100 -0.0365 2.9667 0.2552 A889 C161 -1.0348261 2.5280129 0.2324966 0.4865722
101 1.3869 -2.0095 -0.8972 A86 C283 1.7808158 -0.9337703 -0.8105779 0.5187559
102 -0.6677 0.8863 -0.4552 A580 C153 -1.0030224 1.4214296 -0.8194117 0.4115546
103 0.8018 1.5204 0.9597 A728 C307 1.5763323 1.3461567 0.8384533 0.3566741
104 0.1104 1.7654 -0.9729 A781 C153 -1.0030224 1.4214296 -0.8194117 0.5369604
105 -1.7521 -1.6907 0.9005 A135 C35 -2.1242339 -0.9715258 0.7096658 0.4273808
106 -0.6146 1.0906 -0.6636 A635 C153 -1.0030224 1.4214296 -0.8194117 0.2916879
107 1.0107 0.3118 0.3349 A471 C249 1.0778184 -0.1223857 0.3159433 0.1734203
108 2.5043 0.9402 0.7378 A639 C307 1.5763323 1.3461567 0.8384533 0.4781926
109 2.2697 -0.3642 0.9543 A300 C340 2.6666782 -0.1317501 0.4147766 0.3896505
110 0.9096 0.4156 0.0115 A486 C249 1.0778184 -0.1223857 0.3159433 0.3368824
111 -0.5917 2.7188 -0.6227 A873 C161 -1.0348261 2.5280129 0.2324966 0.4963699
112 0.3515 0.9388 -0.0696 A585 C240 0.6514734 1.0103327 -0.5200428 0.2739830
113 0.0061 -1.0002 -0.0184 A302 C173 0.0988042 -1.1378972 -0.4550217 0.2223410
114 -1.0821 0.8067 -0.7597 A613 C153 -1.0030224 1.4214296 -0.8194117 0.2511729
115 2.8004 0.9339 -0.3059 A672 C330 2.2163811 0.9452850 -0.6878786 0.3257942
116 1.3163 1.9599 -0.9326 A783 C293 0.7212818 2.5491813 -0.2728450 0.6146848
117 -1.4733 -1.3145 0.9997 A221 C35 -2.1242339 -0.9715258 0.7096658 0.4279808
118 -0.2122 -2.6705 -0.7342 A60 C193 0.8084450 -2.5111158 -0.3496015 0.5215426
119 -2.6239 -0.2407 -0.7726 A446 C51 -2.2655670 0.0495469 -0.7279562 0.2310746
120 -1.0119 1.5636 -0.9905 A741 C153 -1.0030224 1.4214296 -0.8194117 0.1073788
121 -2.7687 0.9605 0.3660 A742 C76 -2.1614253 1.1751861 0.5762133 0.3440580
122 -1.5127 -0.2305 -0.8828 A449 C67 -1.4757775 -1.2955060 -0.8486053 0.3787077
123 -1.7557 -1.3185 -0.9807 A171 C67 -1.4757775 -1.2955060 -0.8486053 0.1450037
124 2.3165 1.2526 -0.7738 A709 C330 2.2163811 0.9452850 -0.6878786 0.1644518
125 -0.2336 0.9724 0.0084 A595 C197 -0.1968420 1.3899623 0.6958189 0.3805797
126 -0.5609 2.5334 -0.8039 A855 C161 -1.0348261 2.5280129 0.2324966 0.5052366
127 -0.5395 -0.8478 0.0993 A341 C127 -1.0331492 0.0055723 0.1700299 0.4725838
128 0.6976 2.5253 0.7847 A831 C293 0.7212818 2.5491813 -0.2728450 0.3683694
129 -2.0161 1.8310 -0.6904 A822 C153 -1.0030224 1.4214296 -0.8194117 0.5172199
130 -0.7381 0.6864 -0.1256 A550 C127 -1.0331492 0.0055723 0.1700299 0.4238356
131 -1.6351 0.6072 -0.9667 A555 C51 -2.2655670 0.0495469 -0.7279562 0.4756213
132 -1.3938 -1.5617 0.9956 A181 C144 -0.2145520 -1.5164273 0.7835631 0.4788525
133 -2.1713 -0.7846 0.9512 A366 C35 -2.1242339 -0.9715258 0.7096658 0.1585087
134 2.9855 -0.2779 -0.0572 A256 C340 2.6666782 -0.1317501 0.4147766 0.3123161
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523 2.5715 1.3977 0.3755 A725 C307 1.5763323 1.3461567 0.8384533 0.5032214
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526 1.4018 2.6324 0.1869 A871 C293 0.7212818 2.5491813 -0.2728450 0.4078273
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531 -0.1033 1.0084 -0.1649 A603 C240 0.6514734 1.0103327 -0.5200428 0.3706163
532 1.0450 -0.8073 0.7337 A319 C249 1.0778184 -0.1223857 0.3159433 0.3784965
533 1.0334 0.1543 -0.2962 A441 C249 1.0778184 -0.1223857 0.3159433 0.3110824
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536 -2.2817 -0.4067 -0.9482 A371 C51 -2.2655670 0.0495469 -0.7279562 0.2308746
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541 -0.3747 0.9601 0.2456 A597 C197 -0.1968420 1.3899623 0.6958189 0.3526464
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545 1.9179 1.3814 0.9315 A702 C307 1.5763323 1.3461567 0.8384533 0.1566192
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550 0.7288 0.7442 0.2855 A549 C240 0.6514734 1.0103327 -0.5200428 0.3830007
551 2.2831 0.0150 -0.9591 A401 C330 2.2163811 0.9452850 -0.6878786 0.4227417
552 -2.5635 1.5580 -0.0168 A866 C76 -2.1614253 1.1751861 0.5762133 0.4593006
553 0.9634 0.2999 0.1341 A487 C249 1.0778184 -0.1223857 0.3159433 0.2395158
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556 -1.8007 0.5261 -0.9923 A599 C51 -2.2655670 0.0495469 -0.7279562 0.4019213
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558 0.0628 2.5775 0.8159 A852 C197 -0.1968420 1.3899623 0.6958189 0.5224203
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562 -0.1052 1.0548 -0.3414 A606 C240 0.6514734 1.0103327 -0.5200428 0.3265945
563 1.6380 1.3371 0.9934 A679 C307 1.5763323 1.3461567 0.8384533 0.0752237
564 1.7682 0.6180 -0.9919 A511 C330 2.2163811 0.9452850 -0.6878786 0.3598291
565 -1.2428 -2.7287 0.0575 A34 C46 -1.0353834 -2.5511392 0.0634906 0.1303227
566 -0.0286 1.0078 0.1275 A601 C197 -0.1968420 1.3899623 0.6958189 0.3729077
567 0.8838 -1.0687 0.7899 A270 C264 1.4868170 -1.5149655 0.8303546 0.3632457
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569 -1.9040 1.6712 -0.8459 A822 C153 -1.0030224 1.4214296 -0.8194117 0.3924121
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578 -0.1932 -1.2543 -0.6824 A246 C173 0.0988042 -1.1378972 -0.4550217 0.2119284
579 0.8000 -0.9510 -0.6531 A271 C173 0.0988042 -1.1378972 -0.4550217 0.3620571
580 1.4962 -0.2775 -0.8782 A397 C283 1.7808158 -0.9337703 -0.8105779 0.3361694
581 -2.6190 0.9850 -0.6025 A763 C51 -2.2655670 0.0495469 -0.7279562 0.4714474
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605 2.5591 1.5461 -0.1419 A767 C330 2.2163811 0.9452850 -0.6878786 0.4965042
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612 0.9753 -0.5500 -0.4744 A361 C249 1.0778184 -0.1223857 0.3159433 0.4401587
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616 1.0557 -0.9141 0.7973 A272 C264 1.4868170 -1.5149655 0.8303546 0.3550124
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621 1.3409 0.7729 0.8919 A543 C307 1.5763323 1.3461567 0.8384533 0.2873786
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623 -2.4216 0.1088 -0.9056 A537 C51 -2.2655670 0.0495469 -0.7279562 0.1309766
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629 0.4118 1.2926 -0.7656 A643 C240 0.6514734 1.0103327 -0.5200428 0.2558327
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631 1.9110 -0.7038 0.9993 A239 C264 1.4868170 -1.5149655 0.8303546 0.4680980
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637 1.8595 0.3699 0.9946 A470 C307 1.5763323 1.3461567 0.8384533 0.4718570
638 2.5982 1.3225 -0.4025 A713 C330 2.2163811 0.9452850 -0.6878786 0.3481375
639 0.4991 2.8600 -0.4292 A862 C293 0.7212818 2.5491813 -0.2728450 0.2297852
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647 0.2582 -0.9978 -0.2456 A286 C173 0.0988042 -1.1378972 -0.4550217 0.1696382
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666 1.0541 0.5678 0.5964 A524 C249 1.0778184 -0.1223857 0.3159433 0.3314536
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684 1.5240 -2.5769 -0.1107 A6 C193 0.8084450 -2.5111158 -0.3496015 0.3400802
685 0.7573 0.6729 -0.1608 A540 C240 0.6514734 1.0103327 -0.5200428 0.2675007
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689 0.9760 -0.2515 0.1252 A409 C249 1.0778184 -0.1223857 0.3159433 0.1405587
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701 1.2057 0.1009 -0.6130 A456 C249 1.0778184 -0.1223857 0.3159433 0.4267035
702 0.8516 0.8151 -0.5707 A541 C240 0.6514734 1.0103327 -0.5200428 0.1486722
703 0.6533 -1.2453 0.8047 A211 C264 1.4868170 -1.5149655 0.8303546 0.3762790
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712 -1.8617 -2.3291 0.1904 A41 C46 -1.0353834 -2.5511392 0.0634906 0.3917551
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715 1.2965 -1.0996 -0.9539 A206 C283 1.7808158 -0.9337703 -0.8105779 0.2644892
716 -1.2545 1.2382 0.9714 A730 C76 -2.1614253 1.1751861 0.5762133 0.4550420
717 1.1090 -0.6640 -0.7068 A334 C283 1.7808158 -0.9337703 -0.8105779 0.3484546
718 -0.4117 -1.0055 0.4070 A310 C144 -0.2145520 -1.5164273 0.7835631 0.3615461
719 0.7448 0.6819 0.1398 A549 C240 0.6514734 1.0103327 -0.5200428 0.3605340
720 1.7770 0.1665 0.9766 A423 C307 1.5763323 1.3461567 0.8384533 0.5061570
721 1.6482 -0.4822 -0.9592 A348 C283 1.7808158 -0.9337703 -0.8105779 0.2442694
722 1.0099 0.0319 0.1441 A430 C249 1.0778184 -0.1223857 0.3159433 0.1313491
723 -1.3254 -0.4557 -0.8012 A418 C67 -1.4757775 -1.2955060 -0.8486053 0.3458630
724 -2.2024 1.4958 0.7492 A819 C76 -2.1614253 1.1751861 0.5762133 0.1781918
725 2.2684 1.9133 0.2527 A826 C307 1.5763323 1.3461567 0.8384533 0.6149881
726 0.9630 0.4460 -0.3447 A501 C240 0.6514734 1.0103327 -0.5200428 0.3504007
727 -1.2437 2.5547 -0.5405 A879 C161 -1.0348261 2.5280129 0.2324966 0.3361859
728 -1.2079 1.1582 0.9452 A687 C76 -2.1614253 1.1751861 0.5762133 0.4464994
729 -0.5154 -0.8570 0.0111 A341 C173 0.0988042 -1.1378972 -0.4550217 0.4537410
730 -0.5493 -1.0761 -0.6108 A298 C173 0.0988042 -1.1378972 -0.4550217 0.2885599
731 2.0058 2.0390 0.5099 A802 C307 1.5763323 1.3461567 0.8384533 0.4836214
732 -1.0054 0.1001 -0.1434 A482 C127 -1.0331492 0.0055723 0.1700299 0.1452356
733 -0.6455 -0.8080 -0.2591 A343 C173 0.0988042 -1.1378972 -0.4550217 0.4233744
734 -1.0379 -0.2978 0.3913 A434 C127 -1.0331492 0.0055723 0.1700299 0.1764644
735 -1.2209 -1.5068 0.9982 A167 C144 -0.2145520 -1.5164273 0.7835631 0.4102041
736 0.8193 1.1242 -0.7932 A618 C240 0.6514734 1.0103327 -0.5200428 0.1849504
737 1.8490 0.9128 0.9981 A634 C307 1.5763323 1.3461567 0.8384533 0.2885570
738 -0.3372 -2.4405 -0.8860 A60 C193 0.8084450 -2.5111158 -0.3496015 0.5842198
739 -0.2006 1.0459 -0.3546 A603 C240 0.6514734 1.0103327 -0.5200428 0.3510278
740 0.1127 -1.8314 0.9863 A148 C144 -0.2145520 -1.5164273 0.7835631 0.2816539
741 1.8791 2.2742 -0.3120 A854 C293 0.7212818 2.5491813 -0.2728450 0.4906515
742 2.5492 -1.5473 -0.1886 A85 C283 1.7808158 -0.9337703 -0.8105779 0.6679639
743 -1.1907 2.6973 0.3169 A895 C161 -1.0348261 2.5280129 0.2324966 0.1365215
744 1.5662 2.5045 -0.3002 A869 C293 0.7212818 2.5491813 -0.2728450 0.3056515
745 2.4730 -0.2776 0.8725 A382 C340 2.6666782 -0.1317501 0.4147766 0.2657505
746 1.1493 -0.1922 -0.5506 A402 C249 1.0778184 -0.1223857 0.3159433 0.3359464
747 0.5183 1.8512 -0.9970 A748 C240 0.6514734 1.0103327 -0.5200428 0.4836660
748 -0.8446 -0.9020 0.6448 A316 C144 -0.2145520 -1.5164273 0.7835631 0.4610795
749 2.4197 -1.7142 -0.2610 A50 C283 1.7808158 -0.9337703 -0.8105779 0.6562973
750 -2.1413 -1.4033 0.8284 A165 C35 -2.1242339 -0.9715258 0.7096658 0.1891915
751 -2.5379 0.6876 -0.7771 A723 C51 -2.2655670 0.0495469 -0.7279562 0.3198433
752 0.7839 -2.8918 0.0879 A4 C193 0.8084450 -2.5111158 -0.3496015 0.2809102
753 0.3511 2.4340 -0.8883 A838 C293 0.7212818 2.5491813 -0.2728450 0.3669393
754 0.1350 -1.2130 -0.6263 A248 C173 0.0988042 -1.1378972 -0.4550217 0.0941923
755 0.3113 2.8561 -0.4878 A885 C293 0.7212818 2.5491813 -0.2728450 0.3106185
756 0.2665 -1.7114 0.9634 A156 C144 -0.2145520 -1.5164273 0.7835631 0.2852872
757 1.7459 0.8381 0.9980 A559 C307 1.5763323 1.3461567 0.8384533 0.2790570
758 -1.1116 -0.1985 -0.4916 A445 C127 -1.0331492 0.0055723 0.1700299 0.3147177
759 1.1286 0.6666 -0.7245 A545 C240 0.6514734 1.0103327 -0.5200428 0.3417722
760 -0.5464 2.5419 0.8000 A835 C161 -1.0348261 2.5280129 0.2324966 0.3566055
761 -1.0319 0.1283 -0.2795 A482 C127 -1.0331492 0.0055723 0.1700299 0.1911689
762 0.6609 -1.1062 -0.7027 A253 C173 0.0988042 -1.1378972 -0.4550217 0.2804904
763 -2.0774 0.3179 -0.9948 A528 C51 -2.2655670 0.0495469 -0.7279562 0.2411213
764 -2.4430 -1.0993 0.7342 A224 C35 -2.1242339 -0.9715258 0.7096658 0.1570248
765 0.7547 -0.6989 -0.2374 A356 C173 0.0988042 -1.1378972 -0.4550217 0.4375049
766 -2.9900 0.1337 -0.1185 A525 C51 -2.2655670 0.0495469 -0.7279562 0.4726808
767 1.1471 1.1879 0.9373 A660 C307 1.5763323 1.3461567 0.8384533 0.2287786
768 -2.6938 0.1317 -0.7170 A546 C51 -2.2655670 0.0495469 -0.7279562 0.1737808
769 0.6380 1.4402 0.9053 A681 C197 -0.1968420 1.3899623 0.6958189 0.3648536
770 -0.5219 -2.7379 -0.6167 A47 C46 -1.0353834 -2.5511392 0.0634906 0.4601449
771 -1.2095 -0.9892 0.8992 A295 C35 -2.1242339 -0.9715258 0.7096658 0.3739808
772 1.9367 2.2905 -0.0298 A860 B6 1.8250643 2.3704714 0.0338500 0.0850857
773 -0.2198 -2.9918 -0.0149 A14 C46 -1.0353834 -2.5511392 0.0634906 0.4448783
774 -2.8806 -0.7633 0.1991 A368 C35 -2.1242339 -0.9715258 0.7096658 0.4917192
775 1.2701 -1.4700 -0.9984 A151 C283 1.7808158 -0.9337703 -0.8105779 0.4115892
776 1.7900 -1.6445 -0.9025 A106 C283 1.7808158 -0.9337703 -0.8105779 0.2706120
777 0.2788 0.9731 0.1561 A586 C240 0.6514734 1.0103327 -0.5200428 0.3620163
778 -0.6387 0.9864 -0.5653 A623 C153 -1.0030224 1.4214296 -0.8194117 0.3511546
779 0.2667 1.2938 0.7341 A663 C197 -0.1968420 1.3899623 0.6958189 0.1993285
780 2.8609 0.1105 -0.5052 A442 C340 2.6666782 -0.1317501 0.4147766 0.4521495
781 -1.1411 -2.2151 -0.8707 A77 C67 -1.4757775 -1.2955060 -0.8486053 0.4254554
782 0.2645 -1.7999 0.9835 A148 C144 -0.2145520 -1.5164273 0.7835631 0.3208205
783 -1.4462 1.0908 -0.9821 A706 C153 -1.0030224 1.4214296 -0.8194117 0.3121651
784 -0.7567 2.0248 0.9869 A769 C197 -0.1968420 1.3899623 0.6958189 0.4952589
785 -1.1822 0.6553 0.7613 A566 C127 -1.0331492 0.0055723 0.1700299 0.4633495
786 -1.8790 -1.9855 -0.6795 A93 C67 -1.4757775 -1.2955060 -0.8486053 0.4207739
787 -0.0835 1.0285 -0.2503 A606 C240 0.6514734 1.0103327 -0.5200428 0.3409612
788 -0.3886 2.6981 0.6877 A865 C161 -1.0348261 2.5280129 0.2324966 0.4238389
789 2.0591 2.1030 -0.3322 A833 C330 2.2163811 0.9452850 -0.6878786 0.5568916
790 -1.4258 2.0107 -0.8854 A816 C153 -1.0030224 1.4214296 -0.8194117 0.3593454
791 2.6746 0.0024 -0.7382 A420 C340 2.6666782 -0.1317501 0.4147766 0.4316828
792 1.2197 -0.6145 -0.7732 A334 C283 1.7808158 -0.9337703 -0.8105779 0.3059213
793 1.1799 2.7402 -0.1813 A877 C293 0.7212818 2.5491813 -0.2728450 0.2470607
794 1.1567 2.6821 -0.3899 A877 C293 0.7212818 2.5491813 -0.2728450 0.2284640
795 -1.4169 -0.9925 0.9628 A251 C35 -2.1242339 -0.9715258 0.7096658 0.3271474
796 -2.2685 1.4878 0.7013 A819 C76 -2.1614253 1.1751861 0.5762133 0.1815918
797 1.0089 -0.1229 0.1800 A430 C249 1.0778184 -0.1223857 0.3159433 0.0684587
798 -2.3979 0.7662 0.8558 A718 C76 -2.1614253 1.1751861 0.5762133 0.3083492
799 -1.4404 1.5012 -0.9968 A759 C153 -1.0030224 1.4214296 -0.8194117 0.2315121
800 0.1252 1.0987 0.4477 A621 C197 -0.1968420 1.3899623 0.6958189 0.2871411
801 -2.9051 0.6662 -0.1963 A734 C51 -2.2655670 0.0495469 -0.7279562 0.5959474
802 -2.8443 -0.4389 -0.4787 A379 C51 -2.2655670 0.0495469 -0.7279562 0.4388120
803 -0.9229 -1.0817 -0.8160 A275 C67 -1.4757775 -1.2955060 -0.8486053 0.2664296
804 -0.8605 0.6766 -0.4246 A551 C153 -1.0030224 1.4214296 -0.8194117 0.4273879
805 -0.9142 0.5823 -0.4009 A551 C127 -1.0331492 0.0055723 0.1700299 0.4222023
806 -0.9733 -1.6410 0.9958 A170 C144 -0.2145520 -1.5164273 0.7835631 0.3651859
807 -1.5423 -0.5428 -0.9310 A384 C67 -1.4757775 -1.2955060 -0.8486053 0.3005411
808 0.5563 -2.1322 0.9791 A110 C144 -0.2145520 -1.5164273 0.7835631 0.5273872
809 -1.5315 1.4850 -0.9911 A759 C153 -1.0030224 1.4214296 -0.8194117 0.2545788
810 -0.1740 -0.9957 0.1463 A306 C173 0.0988042 -1.1378972 -0.4550217 0.3387744
811 -0.9558 -0.7741 0.6380 A362 C127 -1.0331492 0.0055723 0.1700299 0.4416638
812 -1.1562 0.1925 0.5608 A492 C127 -1.0331492 0.0055723 0.1700299 0.2335829
813 -0.8363 0.5737 -0.1675 A550 C127 -1.0331492 0.0055723 0.1700299 0.3675023
814 -2.3997 0.2924 -0.9087 A638 C51 -2.2655670 0.0495469 -0.7279562 0.1859100
815 -1.3016 0.1869 -0.7285 A496 C51 -2.2655670 0.0495469 -0.7279562 0.3672880
816 1.1794 1.3855 -0.9836 A666 C240 0.6514734 1.0103327 -0.5200428 0.4555504
817 2.9450 -0.5316 0.1218 A205 C340 2.6666782 -0.1317501 0.4147766 0.3237161
818 1.1667 -2.7515 -0.1502 A3 C193 0.8084450 -2.5111158 -0.3496015 0.2660135
819 -1.8635 -1.8478 -0.7812 A93 C67 -1.4757775 -1.2955060 -0.8486053 0.3358073
820 -2.5402 0.4521 -0.8145 A671 C51 -2.2655670 0.0495469 -0.7279562 0.2545766
821 -1.3961 -0.0252 0.7973 A475 C127 -1.0331492 0.0055723 0.1700299 0.3403311
822 1.6055 0.5331 -0.9513 A519 C330 2.2163811 0.9452850 -0.6878786 0.4288291
823 0.7855 1.3154 -0.8838 A646 C240 0.6514734 1.0103327 -0.5200428 0.2676170
824 1.6173 1.3443 0.9947 A679 C307 1.5763323 1.3461567 0.8384533 0.0663570
825 2.1857 0.1329 0.9818 A470 C340 2.6666782 -0.1317501 0.4147766 0.4375506
826 -0.4027 -2.9450 0.2332 A20 C46 -1.0353834 -2.5511392 0.0634906 0.3987512
827 0.9464 -0.8410 0.6792 A319 C249 1.0778184 -0.1223857 0.3159433 0.4044298
828 0.9591 0.8549 -0.6989 A545 C240 0.6514734 1.0103327 -0.5200428 0.2139722
829 0.4264 -2.8695 0.4338 A24 C193 0.8084450 -2.5111158 -0.3496015 0.5079436
830 0.9074 0.4300 -0.0909 A505 C249 1.0778184 -0.1223857 0.3159433 0.3765491
831 1.0656 -2.5540 -0.6412 A38 C193 0.8084450 -2.5111158 -0.3496015 0.1972125
832 0.9906 1.2340 -0.9086 A655 C240 0.6514734 1.0103327 -0.5200428 0.3171170
833 0.9983 -0.0877 0.0661 A430 C249 1.0778184 -0.1223857 0.3159433 0.1213491
834 2.0065 1.8611 0.6762 A802 C307 1.5763323 1.3461567 0.8384533 0.3691214
835 1.8496 -0.3184 0.9924 A345 C340 2.6666782 -0.1317501 0.4147766 0.5271172
836 -0.7873 -2.4682 -0.8069 A71 C46 -1.0353834 -2.5511392 0.0634906 0.4004711
837 2.2972 1.0007 -0.8627 A664 C330 2.2163811 0.9452850 -0.6878786 0.1036851
838 0.4212 1.6303 -0.9487 A699 C240 0.6514734 1.0103327 -0.5200428 0.4262993
839 1.8060 -1.3900 -0.9603 A146 C283 1.7808158 -0.9337703 -0.8105779 0.2103787
840 -0.0192 1.6302 -0.9292 A726 C153 -1.0030224 1.4214296 -0.8194117 0.4341271
841 -2.8836 0.5040 0.3743 A683 C76 -2.1614253 1.1751861 0.5762133 0.5317580
842 0.9613 1.4100 0.9560 A675 C307 1.5763323 1.3461567 0.8384533 0.2654741
843 -0.3794 -0.9701 -0.2857 A304 C173 0.0988042 -1.1378972 -0.4550217 0.2717744
844 -1.3547 -2.5256 -0.5001 A61 C46 -1.0353834 -2.5511392 0.0634906 0.3028155
845 -1.0246 -0.2662 -0.3374 A428 C127 -1.0331492 0.0055723 0.1700299 0.2625838
846 -2.4370 -1.3451 -0.6212 A131 C67 -1.4757775 -1.2955060 -0.8486053 0.4127406
847 2.4208 1.7307 -0.2186 A767 C330 2.2163811 0.9452850 -0.6878786 0.4863709
848 0.6859 -1.4071 0.9006 A211 C264 1.4868170 -1.5149655 0.8303546 0.3263426
849 1.0868 0.7894 0.7541 A579 C307 1.5763323 1.3461567 0.8384533 0.3768808
850 0.1460 1.1172 -0.4871 A615 C240 0.6514734 1.0103327 -0.5200428 0.2150945
851 1.0309 0.1301 -0.2770 A441 C249 1.0778184 -0.1223857 0.3159433 0.2974491
852 0.9375 0.5325 0.3876 A507 C249 1.0778184 -0.1223857 0.3159433 0.2889536
853 -0.2478 1.0043 -0.2601 A603 C240 0.6514734 1.0103327 -0.5200428 0.3884163
854 0.5722 -0.8538 -0.2340 A328 C173 0.0988042 -1.1378972 -0.4550217 0.3261716
855 -2.9705 0.3954 -0.0807 A669 C51 -2.2655670 0.0495469 -0.7279562 0.5660141
856 -0.6880 0.7644 0.2366 A567 C127 -1.0331492 0.0055723 0.1700299 0.3901823
857 -1.2596 -0.7977 -0.8608 A337 C67 -1.4757775 -1.2955060 -0.8486053 0.2420594
858 1.7439 1.2310 0.9909 A679 C307 1.5763323 1.3461567 0.8384533 0.1450570
859 -1.0599 -2.5859 -0.6070 A49 C46 -1.0353834 -2.5511392 0.0634906 0.2432560
860 0.0909 -1.6516 0.9383 A148 C144 -0.2145520 -1.5164273 0.7835631 0.1984539
861 2.0308 -0.0006 -0.9995 A455 C283 1.7808158 -0.9337703 -0.8105779 0.4573589
862 -0.3798 -1.1834 -0.6533 A254 C173 0.0988042 -1.1378972 -0.4550217 0.2407951
863 -0.4064 1.9079 0.9988 A774 C197 -0.1968420 1.3899623 0.6958189 0.3434923
864 -2.0250 -0.6491 0.9920 A376 C35 -2.1242339 -0.9715258 0.7096658 0.2346646
865 1.8123 -2.3046 0.3630 A11 C264 1.4868170 -1.5149655 0.8303546 0.5274907
866 -1.4801 0.6117 0.9172 A592 C76 -2.1614253 1.1751861 0.5762133 0.5285994
867 0.5761 -1.5471 0.9371 A175 C144 -0.2145520 -1.5164273 0.7835631 0.3249539
868 -1.5165 0.5741 -0.9256 A565 C153 -1.0030224 1.4214296 -0.8194117 0.4889985
869 1.5637 0.3822 -0.9207 A477 C330 2.2163811 0.9452850 -0.6878786 0.4828625
870 1.7442 -2.3123 -0.4433 A22 C193 0.8084450 -2.5111158 -0.3496015 0.4094231
871 1.4146 2.6352 -0.1346 A869 C293 0.7212818 2.5491813 -0.2728450 0.3058607
872 2.6176 0.1761 -0.7818 A465 C330 2.2163811 0.9452850 -0.6878786 0.4214417
873 -2.5544 -1.5404 -0.1838 A129 C35 -2.1242339 -0.9715258 0.7096658 0.6308354
874 0.4846 0.9947 0.4490 A587 C240 0.6514734 1.0103327 -0.5200428 0.3838496
875 -2.3887 0.7546 -0.8631 A686 C51 -2.2655670 0.0495469 -0.7279562 0.3211100
876 -0.3608 -1.0169 0.3896 A315 C144 -0.2145520 -1.5164273 0.7835631 0.3465795
877 -1.4633 -0.3968 0.8751 A396 C35 -2.1242339 -0.9715258 0.7096658 0.4670313
878 -1.0630 0.9016 0.7954 A628 C197 -0.1968420 1.3899623 0.6958189 0.4847005
879 -1.6883 1.6248 -0.9393 A759 C153 -1.0030224 1.4214296 -0.8194117 0.3361788
880 -1.9880 0.3898 0.9997 A604 C76 -2.1614253 1.1751861 0.5762133 0.4607660
881 -2.2371 -0.7257 0.9360 A366 C35 -2.1242339 -0.9715258 0.7096658 0.1950087
882 -2.3828 -1.6542 -0.4344 A96 C67 -1.4757775 -1.2955060 -0.8486053 0.5599739
883 2.8383 -0.6031 -0.4325 A204 C340 2.6666782 -0.1317501 0.4147766 0.4967494
884 -2.9896 -0.0333 -0.1429 A525 C51 -2.2655670 0.0495469 -0.7279562 0.4639787
885 -1.2532 1.3674 0.9894 A730 C197 -0.1968420 1.3899623 0.6958189 0.4575005
886 1.4762 0.8821 0.9599 A637 C307 1.5763323 1.3461567 0.8384533 0.2285453
887 0.4488 -2.2097 0.9670 A110 C144 -0.2145520 -1.5164273 0.7835631 0.5133539
888 -0.3277 1.2964 0.7488 A658 C197 -0.1968420 1.3899623 0.6958189 0.0924671
889 -0.8363 0.6378 0.3175 A544 C127 -1.0331492 0.0055723 0.1700299 0.3255156
890 -2.7794 1.1095 -0.1211 A778 C76 -2.1614253 1.1751861 0.5762133 0.4603247
891 -1.4982 -0.6681 -0.9331 A384 C67 -1.4757775 -1.2955060 -0.8486053 0.2447744
892 2.0064 0.2239 -0.9998 A455 C330 2.2163811 0.9452850 -0.6878786 0.4144291
893 1.4153 -0.9791 0.9603 A236 C264 1.4868170 -1.5149655 0.8303546 0.2457760
894 -2.0325 1.9894 0.5363 A857 C76 -2.1614253 1.1751861 0.5762133 0.3276842
895 -1.3811 2.0926 -0.8618 A816 C153 -1.0030224 1.4214296 -0.8194117 0.3638788
896 -0.6098 1.1700 0.7326 A684 C197 -0.1968420 1.3899623 0.6958189 0.2232338
897 -2.4952 -1.1168 -0.6795 A184 C67 -1.4757775 -1.2955060 -0.8486053 0.4557446
898 0.0304 -1.9895 -0.9999 A125 C173 0.0988042 -1.1378972 -0.4550217 0.4882951
899 1.3934 0.7061 0.8990 A543 C307 1.5763323 1.3461567 0.8384533 0.2945119
900 0.2342 -2.0368 0.9987 A117 C144 -0.2145520 -1.5164273 0.7835631 0.3947539
901 0.8772 0.9646 0.7179 A594 C307 1.5763323 1.3461567 0.8384533 0.4004141
902 -2.6869 1.3327 0.0373 A828 C76 -2.1614253 1.1751861 0.5762133 0.4073006
903 2.0485 0.1805 -0.9984 A455 C330 2.2163811 0.9452850 -0.6878786 0.4143958
904 -0.3375 1.2580 -0.7166 A642 C153 -1.0030224 1.4214296 -0.8194117 0.3105879
905 -0.6048 -2.1226 0.9783 A112 C144 -0.2145520 -1.5164273 0.7835631 0.3970525
906 -1.0933 1.7274 0.9990 A762 C197 -0.1968420 1.3899623 0.6958189 0.5123589
907 0.1438 1.9673 -0.9996 A781 C153 -1.0030224 1.4214296 -0.8194117 0.6242937
908 -2.4747 0.8327 -0.7916 A723 C51 -2.2655670 0.0495469 -0.7279562 0.3519766
909 1.1076 -2.5960 0.5690 A16 C193 0.8084450 -2.5111158 -0.3496015 0.4342135
910 0.5589 1.2429 0.7707 A644 C197 -0.1968420 1.3899623 0.6958189 0.3258952
911 -1.0930 -0.6141 0.6656 A395 C127 -1.0331492 0.0055723 0.1700299 0.3916977
912 1.3891 0.2267 0.8056 A490 C249 1.0778184 -0.1223857 0.3159433 0.3833413
913 -1.5499 -2.2572 -0.6747 A61 C67 -1.4757775 -1.2955060 -0.8486053 0.4032406
914 2.7089 -1.1527 -0.3299 A124 C283 1.7808158 -0.9337703 -0.8105779 0.5425639
915 1.2016 0.4437 -0.6949 A479 C240 0.6514734 1.0103327 -0.5200428 0.4305388
916 0.7562 -2.5736 0.7309 A56 C193 0.8084450 -2.5111158 -0.3496015 0.3984102
917 1.2063 2.3102 -0.7953 A823 C293 0.7212818 2.5491813 -0.2728450 0.4154848
918 1.2973 0.2384 -0.7323 A474 C249 1.0778184 -0.1223857 0.3159433 0.5428369
919 2.1666 -0.6575 -0.9645 A231 C283 1.7808158 -0.9337703 -0.8105779 0.2719922
920 0.3088 -1.3839 0.8132 A232 C144 -0.2145520 -1.5164273 0.7835631 0.2285054
921 -2.9849 0.2803 -0.0623 A669 C51 -2.2655670 0.0495469 -0.7279562 0.5385808
922 -2.2710 -0.2862 0.9573 A443 C35 -2.1242339 -0.9715258 0.7096658 0.3599087
923 -2.2230 -0.2609 -0.9712 A422 C51 -2.2655670 0.0495469 -0.7279562 0.1987526
924 -1.1654 0.0131 0.5510 A463 C127 -1.0331492 0.0055723 0.1700299 0.1735829
925 2.0881 0.5562 0.9870 A530 C307 1.5763323 1.3461567 0.8384533 0.4834237
926 1.0017 2.8271 -0.0355 A881 C293 0.7212818 2.5491813 -0.2728450 0.2652273
927 2.2241 1.6913 0.6077 A779 C307 1.5763323 1.3461567 0.8384533 0.4078881
928 1.4506 -0.7883 0.9371 A273 C264 1.4868170 -1.5149655 0.8303546 0.2898760
929 -0.2724 1.7306 -0.9687 A755 C153 -1.0030224 1.4214296 -0.8194117 0.3963604
930 0.9006 -1.2808 -0.9008 A233 C283 1.7808158 -0.9337703 -0.8105779 0.4391559
931 -0.6270 0.7808 0.0535 A567 C127 -1.0331492 0.0055723 0.1700299 0.4326356
932 0.0206 1.0314 0.2493 A609 C197 -0.1968420 1.3899623 0.6958189 0.3408411
933 -0.1750 -1.6864 0.9525 A164 C144 -0.2145520 -1.5164273 0.7835631 0.1261539
934 -0.0966 2.6236 0.7803 A852 C197 -0.1968420 1.3899623 0.6958189 0.4727870
935 -2.1915 0.5266 0.9672 A604 C76 -2.1614253 1.1751861 0.5762133 0.3565492
936 0.3985 -2.1350 0.9851 A110 C144 -0.2145520 -1.5164273 0.7835631 0.4777205
937 -0.2908 -2.2615 0.9600 A112 C144 -0.2145520 -1.5164273 0.7835631 0.3325859
938 -0.9419 1.7507 -0.9999 A741 C153 -1.0030224 1.4214296 -0.8194117 0.1902937
939 -1.0571 1.2312 -0.9261 A701 C153 -1.0030224 1.4214296 -0.8194117 0.1169985
940 0.5496 -0.8357 0.0190 A332 C173 0.0988042 -1.1378972 -0.4550217 0.4090049
941 -1.4026 -1.7698 -0.9661 A154 C67 -1.4757775 -1.2955060 -0.8486053 0.2216554
942 -1.7156 -0.9813 -0.9997 A296 C67 -1.4757775 -1.2955060 -0.8486053 0.2350411
943 -1.1728 1.9524 -0.9607 A777 C153 -1.0030224 1.4214296 -0.8194117 0.2806788
944 1.1716 1.3992 -0.9846 A655 C240 0.6514734 1.0103327 -0.5200428 0.4578504
945 1.5494 -0.4753 -0.9253 A348 C283 1.7808158 -0.9337703 -0.8105779 0.2682027
946 0.7918 1.6351 -0.9831 A693 C240 0.6514734 1.0103327 -0.5200428 0.4093837
947 -0.6693 -2.6268 0.7034 A64 C46 -1.0353834 -2.5511392 0.0634906 0.3605512
948 -0.4149 -1.0642 -0.5140 A279 C173 0.0988042 -1.1378972 -0.4550217 0.2154599
949 1.3399 2.5298 0.5057 A839 C293 0.7212818 2.5491813 -0.2728450 0.4721815
950 -1.0850 1.2580 -0.9409 A701 C153 -1.0030224 1.4214296 -0.8194117 0.1222985
951 1.4104 -0.0125 0.8077 A431 C249 1.0778184 -0.1223857 0.3159433 0.3114080
952 -0.1689 1.2276 0.6489 A658 C197 -0.1968420 1.3899623 0.6958189 0.0790744
953 -1.2890 -0.0982 0.7070 A452 C127 -1.0331492 0.0055723 0.1700299 0.2988644
954 0.8588 -1.1465 -0.8234 A233 C173 0.0988042 -1.1378972 -0.4550217 0.3789923
955 0.9992 0.2304 0.2241 A459 C249 1.0778184 -0.1223857 0.3159433 0.1744158
956 -1.0154 0.1009 0.2007 A473 C127 -1.0331492 0.0055723 0.1700299 0.0479156
957 -1.1648 1.8204 0.9869 A762 C161 -1.0348261 2.5280129 0.2324966 0.5306634
958 2.3493 1.0052 -0.8316 A664 C330 2.2163811 0.9452850 -0.6878786 0.1121851
959 -1.6549 -2.0691 0.7604 A105 C35 -2.1242339 -0.9715258 0.7096658 0.5392141
960 -0.5279 0.8633 0.1537 A581 C127 -1.0331492 0.0055723 0.1700299 0.4597689
961 1.6959 -0.4758 0.9711 A305 C264 1.4868170 -1.5149655 0.8303546 0.4629980
962 -0.7860 -1.3001 -0.8769 A218 C67 -1.4757775 -1.2955060 -0.8486053 0.2408887
963 0.4045 0.9328 -0.1822 A572 C240 0.6514734 1.0103327 -0.5200428 0.2207830
964 2.8584 -0.1906 0.5023 A412 C340 2.6666782 -0.1317501 0.4147766 0.1126983
965 -2.1611 -1.8664 -0.5178 A79 C67 -1.4757775 -1.2955060 -0.8486053 0.5290073
966 0.9421 0.3621 -0.1362 A484 C249 1.0778184 -0.1223857 0.3159433 0.3574491
967 0.4663 -2.8048 -0.5374 A30 C193 0.8084450 -2.5111158 -0.3496015 0.2745426
968 0.1256 -1.7880 0.9782 A148 C144 -0.2145520 -1.5164273 0.7835631 0.2687872
969 2.6307 0.3229 0.7596 A489 C340 2.6666782 -0.1317501 0.4147766 0.2784839
970 1.8731 0.7602 -0.9998 A568 C330 2.2163811 0.9452850 -0.6878786 0.2800958
971 -2.7520 -0.0244 0.6590 A476 C35 -2.1242339 -0.9715258 0.7096658 0.5418526
972 2.2400 -1.9569 -0.2249 A50 C283 1.7808158 -0.9337703 -0.8105779 0.6893306
973 0.0754 1.2910 -0.7074 A645 C240 0.6514734 1.0103327 -0.5200428 0.3480327
974 2.5594 1.1582 0.5875 A725 C307 1.5763323 1.3461567 0.8384533 0.4739926
975 0.8845 -0.6161 -0.3869 A361 C173 0.0988042 -1.1378972 -0.4550217 0.4585382
976 -0.7740 -0.6682 -0.2110 A369 C127 -1.0331492 0.0055723 0.1700299 0.4379838
977 -0.8790 1.0695 -0.7880 A653 C153 -1.0030224 1.4214296 -0.8194117 0.1691212
978 -1.2881 0.0288 0.7026 A475 C127 -1.0331492 0.0055723 0.1700299 0.2702495
979 -0.6097 -1.0526 -0.6213 A298 C173 0.0988042 -1.1378972 -0.4550217 0.3200266
980 -1.4135 -2.5462 0.4095 A42 C46 -1.0353834 -2.5511392 0.0634906 0.2430217
981 -1.1947 0.2059 -0.6161 A500 C127 -1.0331492 0.0055723 0.1700299 0.3826695
982 1.8473 -2.1395 0.5626 A53 C264 1.4868170 -1.5149655 0.8303546 0.4175907
983 -1.6748 -1.1306 -0.9998 A214 C67 -1.4757775 -1.2955060 -0.8486053 0.1717077
984 0.9959 -0.8871 0.7457 A319 C264 1.4868170 -1.5149655 0.8303546 0.4011457
985 -0.5795 1.8701 0.9991 A769 C197 -0.1968420 1.3899623 0.6958189 0.3886923
986 -1.6731 2.2443 0.6009 A858 C161 -1.0348261 2.5280129 0.2324966 0.4301301
987 0.0221 1.4389 -0.8278 A690 C153 -1.0030224 1.4214296 -0.8194117 0.3503271
988 0.8051 0.6000 -0.0910 A505 C240 0.6514734 1.0103327 -0.5200428 0.3310007
989 -0.1298 -1.5035 -0.8712 A191 C173 0.0988042 -1.1378972 -0.4550217 0.3367951
990 -0.6356 -2.3483 0.9015 A100 C144 -0.2145520 -1.5164273 0.7835631 0.4569525
991 2.2303 0.8724 -0.9187 A665 C330 2.2163811 0.9452850 -0.6878786 0.1058751
992 1.0429 2.0287 -0.9597 A752 C293 0.7212818 2.5491813 -0.2728450 0.5096515
993 -0.4516 2.4733 -0.8577 A832 C153 -1.0030224 1.4214296 -0.8194117 0.5471937
994 -1.2940 0.9559 -0.9203 A651 C153 -1.0030224 1.4214296 -0.8194117 0.2857985
995 -0.9320 1.2782 0.9084 A705 C197 -0.1968420 1.3899623 0.6958189 0.3531671
996 -0.4005 -1.2427 -0.7197 A254 C173 0.0988042 -1.1378972 -0.4550217 0.2895951
997 0.0619 -2.8141 -0.5798 A25 C193 0.8084450 -2.5111158 -0.3496015 0.4265759
998 -2.3791 -1.7302 -0.3363 A96 C67 -1.4757775 -1.2955060 -0.8486053 0.6167739
999 2.2268 -1.5323 0.7111 A98 C264 1.4868170 -1.5149655 0.8303546 0.2921907
1000 -1.8225 -0.7762 -0.9998 A311 C67 -1.4757775 -1.2955060 -0.8486053 0.3390744

hist(act_pred$diff, breaks = 30, col = "blue", main = "Mean Absolute Difference", xlab = "Difference")
Figure 22: Mean Absolute Difference

Figure 22: Mean Absolute Difference

7 Executive Summary

8 References

  1. Topology Preserving Maps : https://users.ics.aalto.fi/jhollmen/dippa/node9.html

  2. Vector Quantization : https://en.wikipedia.org/wiki/Vector_quantization

  3. K-means : https://en.wikipedia.org/wiki/K-means_clustering

  4. Sammon’s Projection : https://en.wikipedia.org/wiki/Sammon_mapping

  5. Voronoi Tessellations : https://en.wikipedia.org/wiki/Centroidal_Voronoi_tessellation